[2019-05-08] Update: Info on this page is now probably outdated. I might update this later. See: https://webmasters.googleblog.com/2019/05/the-new-evergreen-googlebot.html 🎉
Updated on 2019-01-04
Mozilla/5.0 (compatible; Googlebot/2.1; +http://www.google.com/bot.html)
Mozilla/5.0 (X11; Linux x86_64) AppleWebKit/537.36 (KHTML, like Gecko; Google Search Console) Chrome/41.0.2272.118 Safari/537.36
Mozilla/5.0 (Linux; Android 6.0.1; Nexus 5X Build/MMB29P) AppleWebKit/537.36 (KHTML, like Gecko) Chrome/41.0.2272.96 Mobile Safari/537.36 (compatible; Googlebot/2.1; +http://www.google.com/bot.html)
Mozilla/5.0 (Linux; Android 6.0.1; Nexus 5X Build/MMB29P) AppleWebKit/537.36 (KHTML, like Gecko) Chrome/41.0.2272.96 Mobile Safari/537.36 (compatible; Google Search Console)
This information was fetched using the "Fetch and render" feature of "Fetch as Google" tool from Google Search Console by having it fetch a page that displays the user agent string (navigator.userAgent via JS).
Googlebot uses different user agents depending on if you choose to request a fetch with the "Desktop" option or the "Mobile: Smartphone" option.
The tool also shows two different renderings per fetch and both of those use different user agents. One is titled "This is how Googlebot saw the page" (above: Googlebot UA) and the other is titled "This is how a visitor to your website would have seen the page" (above: Googlebot visitor UA).
In some cases Google uses also other user agent strings. Some information about those can be found here: https://support.google.com/webmasters/answer/1061943. Though that page is missing some details like the version of Chrome used by Googlebot for desktop (which you can see above).
If you're interested in locally testing or debugging a website using a browser that is as close as possible to the one that Googlebot is using for these renders, I suggest you install an old version of Chromium that best matches the Chrome version mentioning in the user agents above.
The Download Chromium page has a section about downloading old builds with instructions on how to find a specific old version.
Following that I could not find a build for the version 41.0.2272.96 used by mobile Googlebot but I could find a ready to use Chromium build for the version 41.0.2272.118 used by desktop Googlebot.
Here are shortcut links to quickly grabbing that specific version for your platform.
Chromium 41.0.2272.118 for:
Just download the appropriate file(s) you need from the linked page. I'm not sure if you might need some of the other files in some case but at least for macOS the only thing I needed was chrome-mac.zip from the "macOS" link above and it seems to work fine in macOS 10.14.2 (Mojave).
I make no guarantees about the correctness of this information.
Googlebot has finally been updated to run on a new version of Chromium (hurray!).
See: https://webmasters.googleblog.com/2019/05/the-new-evergreen-googlebot.html
I also linked this info to the top of the gist.