Anyway, for a brief list in a very rough order of what I consider best, with stuff near the bottom of the list still being at the very least "Kinda decent".
I strongly apologize for not including links and some of the descriptions being vague, but that's halfway intentional(there are better synopses and reviews out there) and halfway because this already took damn ages.
I've marked every show with single-letter tags:
- F - Fanservice
- f - Minor amounts of fanservice
- P - Pandering(e.g. lighter usually less sexual fanservice that doesn't compromise a show's quality much)
- p - minor pandering
- S - Sexual themes(sex, major amounts of nudity, erotic stuff)
- s - sexual themes(dirty humor, minor amounts of nudity)
- V - Violence(shows whose description would mandatorily include the word "violent", e.g. graphic violence or a lot of it)
- v - violence(Either major amounts of non-graphic violence, or minor amounts of slightly graphic violence
- T - Triggers(sexual assault mainly)
- t - triggers(depiction of psychological trauma or "horrifying" stuff, usually in minor amounts)
Many of the shows on the list have multiple seasons, extra OVAs, or movies, I'm leaving finding those as an exercise for the reader.
- Fullmetal Alchemist: Brotherhood - Has action, humor, romance, fascinating worldbuilding, a strong, coherent, interesting plot with philosophical undertones, easily the best shounen ever made, a strong contender for one of the best shows out there.
- Shinsekai Yori (sV)- Follows a group of children growing up in an idyllic postapocalyptic village, a millenium after the manifestation of psychokinetic powers among humans destroyed our civilization. One of the best anime of recent years, if not ever, where things are not as they seem.
- Made in Abyss (sV) - A show with an amazingly fascinating setting, that is way more impactful than it initially seems, a really great ride all around. Some people are uncomfortable because the characters are and look like children and there's some nsfw jokes, but I'd argue none of them are sexualized in any way... but oh well.
- Re:ZERO (Pvt)- A young man gets transported to another world... except that things don't really go very lightly for him. Has a bunch of isekai tropes and elements that "pander", but not to a distasteful degree, no outright fanservice, and it has some genuine character development and catharsis, on top of being somewhat impactful and having an interesting story & setting. Has its ups and downs, but certainly a very worthwhile watch.
- Natsume Yuujinchou - Amazing characters, interesting episodic stories, a very wholesome show that isn't devoid of tension and drama. The first season had a low budget and is slowly paced, but stick with it, it's amazing. Has 6 seasons
- Ghost in the Shell: Stand Alone Complex (s) - I'm bad at describing. It's "just" some really good cyberpunk about an elite crime-handling division. Might start out kinda slow, there's also movies but I've personally liked the series much more. No fanservice, but some nudity and one of the characters tend to wear odd outfits.
- Parasyte (sV) - Aliens invade Earth, by falling from the sky and assimilating with humans. The ideal is to take over the mind, but the MC only has his arm taken over. I've only read the manga, but it was great and the show is likely to rock also.
- No Game No Life (FP) - Siblings with godlike gaming prowess are transported to a world where all conflict is resolved with games by its God. Has beyond abundant fanservice and pandering, but I've somehow still loved the show, and that says something of it, in my book.
- Mushishi - A contemplative kind of show - the pace is slow, there is only one main character, it's episodic, but it's also really interesting. Follows a traveling doctor who deals with matters of "mushi", a very primitive lifeform that most cannot see, in a mythological rural Japan like world.
- Ano hi Mita Hana no Namae wo Bokutachi wa Mada Shiranai - A nice touching story, about a group of friends who moved apart years ago following the death of their friend, who has now suddenly appeared as a ghost visible to the protagonist again.
- Monogatari series (fPv) - The audiovisuals are pure art, and the show itself is a great comedy. Fanservice is somewhat plentiful, but not especially distasteful, and usually funny in and of itself. Mind the confusing watch order.
- Death Note (fv) - A shinigami(death god) drops a notebook that can be used to plan people's deaths down into the human world, and it's picked up by Light Yagami, a very intelligent young man with a strong sense of justice, and comparatively few scruples. Doesn't have overt fanservice, but there's a character who tends to wear skimpy clothes and overall, well, you get the drill. Doesn't really detract from the show IMO.
- Code Geass (fpv) - Mecha show with bishounens. Japan, like most of the world, has been conquered by the Britannian Empire, and is now called Area 11. A young Brittanian man, Lelouch, is given a Geass ability by a mysterious girl with green hair, that lets him issue an absolute order to anyone, once per person, and intends to destroy the Brittanian Empire with its power. Has fanservice, but for both male and female viewers, and it doesn't detract from the show.
- The Tatami Galaxy - For a change, I'll copypaste the synopsis from anime-planet: A man is miserable. Despite all his dreams of a “Rose-Colored Campus Life” filled with raven-haired maidens who dote on him, his social life is going nowhere. He has no girlfriend, his only good friend keeps getting him into trouble, and the circle he joined brings him no joy. So he tries again, and again, reliving his first two years of college life ad nauseum, making different decisions each time, having no recollection that he’s already done this all before. Will the man ever be satisfied with how his life turns out?
- Ojii-san no Lamp - A very short mini-movie about a man having trouble adjusting to the coming times. Honestly made me tear up a bit.
- Angel Beats (p) - A young man goes to afterlife where teenagers wage battle against an angel in order to delay their reincarnation. Comedy, feels, and great music if you enjoy jpop.
- Shouwa Genrou Rakugo Shinjuu - As down-to-earth a premise as it gets, but really solid characters and masterful execution, with an interesting subject(a form of traditional Japanese storytelling, rakugo). Better than it sounds, by a good bit.
- Macross Frontier (pf) - a space opera musical with mecha. Has good jpop, some fanservice, some drama. Watched it a longer time ago, so I cannot remember how good it was very well.
- Mob Psycho 100 - An amusing show about a guy with (very strong) psychic powers, and his growth as a person(while still having a bunch of action). Starts looking a bit like an inconspicious OPM-style shounen, but is actually really solid.
- Psycho-Pass (Vt) - Cyberpunk police in Japan under jurisdiction of the Sybil system that grades all humans' mental states. Not perfect by any means, but very solid nonetheless.
- Shiki (VVVt) - A "vampire" anime, that takes 3/4 of the show to get started for good, but just about might be the only anime out there that's actually horrifying.
- Welcome to the NHK (s) - a great serious/comedy show about a hikkikomori(shut-in) NEET(Not in Education, Employment or Training) guy who meets a girl who says she can change his life. Touches a lot on stuff like porn, masturbation, some perversions, but I would not call any of it "fanservice", personally.
- Kemono no Souja Erin (v) - A surprisingly touching, mature show about a girl who has a passion and a knack for raising large animals. A bit slow-paced.
- Steins;Gate (fp) - A great(if a bit weak at some of the seams) show about a group of teenagers who somehow managed to rig their microwave up to send messages into the past, and only the self-proclaimed mad scientist protagonist maintains memory across timelines as they run into serious trouble with an organization named SERN. No fanservice, slight-to-moderate pandering.
- Eve no Jikan - A near-scifi show about a cafe that has policies to make it impossible to tell androids and humans apart, and a young man who finds it after going through his household android's logs.
- Planetes - A very hard(almost strictly realistic) science-fiction comedy about... orbital trashmen, with some serious geopolitical undertones. A very solid show.
- Nichijou - Really great deadpan comedy. It is about elementary school girls mainly, but there's not a single hint of fanservice and even the moe factor is very toned down. A lot of great, oft absurd, deadpan humor, and a great artstyle.
- Kaze no Tani no Nausicaa - The adventures of a young princess in a world a millenium after an environmental disaster, one of the biggest anime classics out there.
- Redline (fsv) - An scifi movie about the most dangerous and high-profile illegal race in the galaxy. Absolutely amazing visuals, with every frame hand-drawn and highly detailed.
- Fate/Zero - Contemporary fantasy, 7 magi wage war for the "Holy Grail" in a death game ritual. Leagues above the other Fate/ shows, don't knock it just because of that. A bit poorly paced, but well done otherwise.
- The Twelve Kingdoms (v) - A relatively wonkily animated, slowly paced show based on a long series of great novels. What stands out about it is that it has a full-sized fictional mythology, a rich, coherent, fascinating one strongly inspired by the Asian mythology, solid plot, and a great female protagonist. The best way to describe the show might be "it's something of a mythical world historical drama".
- Mirai Nikki (pV) - A shut-in named Yukiteru has a diary and an imaginary friend named Deus-ex-Machina. One of these days, this imaginary friend asks him to play a hunger games type game with 11 other people, where their diaries write themselves a certain distance into the future, to select the next Deus ex Machina. Not /the/ best fiction out there, but it's pretty solid anyway.
- xxxHolic - /Kinda/ like Natsume Yuujinchou - great characters, fascinating world, just a fairly solid episodic show.
- Great Teacher Onizuka (s) - A show about a "delinquent" type teacher, who in spite of his rough demeanor, manages to get through to his problem students, imparting life lessons and helping them when they're in trouble. Might start a bit slow, but it's really solid, trust me. Has some mildly nsfw elements, but.
- Attack on Titan (V) - For the first time in 100 years, humanity is besieged by titans and trembles in fear once again. Well choreographed fights, the plot isn't complete bullshit, and the execution is pretty solid also.
- Haruhi Suzumiya (fp) - A young girl is, without being aware of it, God. Her whims shape reality, and her melancholy threatens its destruction. The story follows Kyon, an ordinary guy, who gets dragged into the middle of it. Fanservice is mostly localized to one character. Also watch the movie, but not the newer series.
- Seirei no Moribito - A spearwoman takes a prince under her wing, to save him from being killed, and together they go through a lot of adventures, eventually growing closer together.
- Durarara!! (P) - A solid comedic show about urban legends, and the people who tangle with them.
- Shirokuma Cafe - I well might be the only one who loved this show about a polar bear running a cafe, with a human helper, a penguin and a panda for regulars. Really heartwarming, relaxing comedy.
- Neko no Ongaeshi - an anime movie from Ghibli, probably my favorite of theirs, where a girl saves a cat, and it turns out he's the prince of a cat kingdom, and they go on an adventure together, just a nice family-friendly story. (tbh you should watch all/most of Ghibli's works)
- Jinrui wa Suitai Shimashita - An odd comedy about a world where the humanity has declined, and fairies who love sweets and despite being stupid, have extremely advanced technology. Not that funny, but has an extremely enjoyable tone and audiovisuals.
- Saiki Kusuo no Ψ-nan - a comedy about the "disastrous" life of a guy born with psychic powers, extraordinarily powerful ones at that, which combined with his weirdo acquaintances, make his life far more eventful than he'd like it to be.
- Eden of the East - A man wakes up in front of the White House with a peculiar smartphone in his hand charged with 8,200,000,000 yen in cash, a gun, and no memory, and subsequently meets a tourist from Japan. Did I mention I'm awful at describing things? The show is pretty nifty.
- Fractale (T) - A bit hard to describe show, about a distant future where people live peacefully under the "Fractale system", that no one understands anymore, because science is pretty much no longer needed. What it does well(setting, characters), it does very well, the rest a bit less. Has some (relatively mild but still) potentially triggering stuff.
- Hai to Gensou no Grimgar (pv) - A group of young people wake up without memory in a world resembling a game. Very climatic, and surprisingly serious. There's fanservice in the bonus episode, and the characters could be done a bit better, but it's still a great watch.
- Spice and Wolf (s) - A traveling merchant meets a pagan deity of harvest whom people have largely forgotten about, and they begin traveling together. Sadly, the 2nd season is far worse than the 1st, and the trading/economy elements aren't as good as you'd hope them to be. I heard that the next un-adapted arc is the best thing in the series, though, if you like it and are up to reading the LNs. Has some nudity, but not fanservice.
- Death Parade (v) - A short, self-contained show about "arbiters" who judge whether human souls should be reincarnated or cast into the void in the afterlife. Has a fairly cool cast, and while I wasn't overly moved by its plot, it was nonetheless a worthwhile watch.
- Samurai Champloo (v) - It's been too long since I watched this one to remember it well, but it's basically samurai x feudal japan x (actually good) hiphop. If that sounds like your kinda thing, you'll love the show.
- Toki wo Kakeru Shoujo - A girl realizes she can jump back in time, what she doesn't realize, is that it ends up having serious consequences. A nice movie.
- Baccano! - A comedic mess involving immortality, train robbery, and a pair of idiots. Might be confusing, since it tells the same events from several perspectives
- Bokurano (t) - A group of teenagers come to be piloting a mecha, and fighting monsters... except there's quite a bit more to it, and it's not very merry. For starters, there are other people piloting those monsters, and the mecha run off the kids' lifeforce...
- Bounen no Xamdou (v) - A kinda odd scifi-fantasy series. I don't remember a lot about it, but it's at least reasonably fun, and somewhat unique, though might be a bit poorly paced and confusing at times.
- Hellsing Ultimate (fV) - Overpowered vampire(s) vs vatican vs nazis, what more do I need to say? Also check out Hellsing Ultimate Abridged for some good laughs.
- Dennou Coil - A story set in a world where augmented reality is the norm, it follows a group of children in a smaller town where said cyberspace is behaving strangely.
- Sora no Woto (p) - Post-apocalyptic slice of life about an all-female unit stationed in a quiet, rural area. Unlike most SoL featuring cute girls, this one is neither boring nor distasteful. If you like this, you would be commiting a crime against yourself by not reading Yokohama Kaidashi Kikou.
- Haibane Renmei - a slightly older(and thus not very well paced) show where a girl wakes up as an angel in an odd walled town, where angels coexist with humans. Sorta slice of life-ish, but not boring.
- Kara no Kyoukai (v) - 8 anime movies with great music and audiovisuals, about a young woman who gained a supernatural ability to kill upon waking from a coma. If you know any nasuverse stuff, you somewhat know what to expect.
- Zankyou no Terror - Hard to make a synopsis. Two teenagers carry out terrorist attacks in Tokyo, and the plot is actually plausible, and the forensics the police do are actually well-researched. Sadly, it's kinda lacking in other aspects: The plot is a bit far-fetched, the police is stupid, in the second half makes no sense, and some other annoyances. Still, a decent enough show.
- Brave Story - A "children's fairy tale" movie, with amazing visuals, pretty interesting.
- Gintama - I have a hard time recommending it, because it's a show that at times can be awful and full of toilet and kindergarten humor and cringy "serious" bits, but it grows on you to a point where you eventually love it. Watch at your own peril.
- Kill la Kill (Fv) - A bit.. hard to describe. Ridiculously over the top comedy about an authoritarian school where (some) students wear uniforms that give them nearly supernatural abilities, and a protagonist wielding a pair of huge scissors. Has a lot of gratuitous near-nudity, but in a hilarious self-aware way.
- Yondemasuyo, Azazel-sun (Sv) - A slapstick-ish comedy about summoning demons. Has plenty of dirty humor, but no fanservice to speak of.
- Puella Magi Madoka Magica - magical girls, except it's not all rose-colored. Kinda too short to be impactful, but the movies are more interesting.
- Sidonia no Kishi (fps) - Science fiction, where humanity escaped Earth on enormous seedships, following attacks by impervious to any conventional Gauna. It's a shounen, with bad-ish 3DCG, wonky fanservice, and tropes bad to a point of being funny, but its scifi elements are thought out and well-developed, and the show itself is interesting. It's a very mixed bag, but I feel that I can recommend it for the parts that it does do well. Keep in mind that the movie is a recap.
- Paprika - The movie that inspired Inception. Has some really nifty animation and music.
- Level E - A very underrated 12 episode comedy(or at least, it made me laugh, in the past) involving aliens
- Ghost Hunt - A shoujo about a paranormal activity private detective kinda thingy, surprisingly decent.
- Black Lagoon (sV) - A Japanese dude visits Southeast Asia carrying a top secret disk for his company, but gets kidnapped by (modern) pirates, and then ends up joining them after his company abandons him. Action, guns, crime. Honestly can't remember much about it, it's been ages since I've last seen it, I remember that second season gets kinda funky. Has some fanservice, but nothing awful.
- Houseki no Kuni - A bit of an odd show, about 28 crystal people living under the guidance of their Teacher, defending themselves from semi-immaterial moon people who periodically attack them, in a distant future where animals mostly seem to not exist anymore, following Phosphophyllite, the youngest of them, unable who is assigned the role of creating an encyclopedia of natural things.
- Katanagatari - An unusually told story about collecting legendary swords created by a legendary swordsmith. I remember little more c.c
- Seto no Hanayome (fP) - A young man is saved from drowning by a mermaid from a mermaid yakuza family, who then reveals to him that he needs to either die for learning their secret, or marry into the family, something that her extremely overprotective badass of a father isn't happy with. A solid romantic comedy.
- Sarai-ya Goyou - A socially awkward samurai comes to Edo seeking to make a name for himself, and ends up involved in what can only be called the criminal underworld. A considerably better show than one would expect.
- Arakawa Under The Bridge - An absurd-ish comedy about a rich man who cannot stand being indebted to anyone whose life is saved by a girl living under a bridge with a bunch of other weirdos, who then begins to live there.
- Assassination Classroom (f)- A fairly decent shounen about a creature that no military can stand up to, opting to teach the worst class at a certain middle school, in both school subject, and assassination, giving them the chance to stop him from destroying the Earth a year since. Has fanservice mostly localized to a single character.
- Kokoro Connect - Five high school students begin to experience randomly swapping their bodies at the whim of a supernatural being called "Heartseed". Kinda bland, but a decent enough show.
- Cowboy Bebop - An anime classic, about a pair of space bounty hunters that I didn't like mainly because its sci-fi elements are really weak and poor pacing.
- Mujin Wakusei Survive - An admittedly average show, more of a kids show than anything else, but thorough 52 episodes, I never really felt bored. It's not flashy, and one of the characters is kinda unbearable, but I've never seen anyone else who has seen it, and I feel like it deserves some more attention than that, it's well done.
- Ergo Proxy - A weird show, some call it "deep", but I don't really consider it such. Great music and unusual story. Slow paced.
- Sayonara Zetsubou Sensei - This show is jampacked with anime and Japanese culture references, puns, jokes, etc. to a point where you likely won't appreciate it unless you've seen a lot of anime, especially early 00s stuff, and have some basic understanding of Japanese culture at the very least. A lot of suicide jokes.
- Boku Dake ga Inai Machi (t) - A guy realizes he has the ability to jump into the past, and strives to use it to prevent a murder that drove him and his childhood friends apart, changing their lives.
- Darker than Black (v) - A show called by some "chinese batman", where ten years ago, mysterious "gates" appeared in Japan, that gave rise to two types of people with supernatural abilities, both coming with severe drawbacks. A fairly decent(if described as kinda bland by some) show. The second season has fanservice, but the first doesn't.
- Rokka no Yuusha (p) - an admittedly kinda bland show, but if you're this far on the list, you're probably desperate enough to enjoy it. And besides, I found it interesting enough. I'd actually gladly watch S2.
- Kore wa Zombie Desu ka (Fp) - the protagonist dies, and gets resurrected as a zombie. He also becomes a magical girl, including skin-showing transformation sequences. There's also all-female secret society vampire ninja clans. Very plentiful fanservice, including from the male protagonist. Watched it ages ago, but found it really funny back then.
- Mawaru Penguindrum (p) - It's... an odd show. More artsy and pretentious than genuinely good, but for one reason or another, I think it just makes the cut for the "anime worth checking out at least" list. For the synopsis, let me just copypaste: The story centres around the lives of three siblings, Takakura Kanba, Takakura Shouma and their terminally ill sister Takakura Himari. Himari passes away after going out on a trip with her brothers. She was however, miraculously saved by a spirit thriving in a penguin-shaped hat. In return for her extended life, the twins must search for an item known as the Penguin Drum with the aid of three penguins which are visible only to the siblings.
- Full Metal Panic (v) & Fumoffu (F) - Fairly bland but well done mecha, recently got a 3rd season. Fumoffu, on the other hand, is one of the best comedies around, IMO. The premise is that Sousuke Sagara, an ex-child soldier who suspects terrorism everywhere and takes everything extremely over-the-top seriously attends public high school in Japan. It's great, though it does have fanservice.
In addition to above, also check out all the Ghibli movies.
I have also thought about mentioning several shows I have not watched yet that I keep my eye on, but well-rated anime that sounds promising so often turns out to be pretty crappy anyway, that I'll not take the risk. I take my recommendations seriously, after all.
I haven't included some noteworthy-ish shows that I disliked here, so it's not a perfectly exhaustive list, but.
As an extra, here are some other shows that might be worth noting that I don't remember, can't make a decision about, or otherwise am in doubt:
- Ghost Hound - kids and spooky supernatural stuff happening
- Sword of the Stranger - samurai action stuff
- Mardock Scramble - kinda disjointed cyberpunk anime movies
- Beelzebub - delinquent comes to be raising a child... from hell, a pretty good comedy, shounen but falling into surprisingly few garbage tropes
- Mnemosyne - weird, probably triggering, gory
- ef - a tale of memories - drama about memory loss, was really moving for me at least.
- Elfen Lied - telekinetic human children are experimented on, and break free
- Pale Cocoon - a man in postapocalyptic future discovers video from the past. very short
- Accel World - decent generic shounen with an obnoxious MC
- Cromartie - weird comedy, kinda bland but with very unique style
- Sakasama no Patema - people live underground, apparently there's an upside down surface
- Noragami - a low-ranking god has trouble staying in business
- Persona 4 The Animation - based on a game, iirc kinda bland