An experimental way to register a CPT where you can have just /%postname%/ rewrite without a prefixed slug. Not fully tested so let me know if something doesn't work. I've wanted to do this from time to time but never really found a good way to do it without some issues.

cpt-without-slug.php

<?php
/*
Plugin Name: CPT Without Slug
Description: A test plugin to register a CPT without slug
Author: David Mårtensson
Version: 0.0.4
Author URI: http://feedmeastraycat.net/
*/

add_action( 'init', 'mycpt_init' );
function mycpt_init() {
  register_post_type( 'mycpt', array(
    'public' => true,
    'rewrite' => array( 'with_front' => false, 'slug' => '/' )
  ) );
}
 
add_action( 'mycpt_rewrite_rules', 'mycpt_remove_rewrite_rules', 10, 1 );
function mycpt_remove_rewrite_rules( $default_rules ) {
  return array();
}
 
add_action( 'pre_get_posts', 'mycpt_pre_get_posts', 11, 1 ); // Should prob run last
function mycpt_pre_get_posts( &$wp_query ) {
  if ( !is_admin() && isset($wp_query->query_vars['name']) && !empty($wp_query->query_vars['name']) ) {
    $wp_query->set( 'post_type', array( 'post', 'mycpt' ) );
  }
}

Issue: Post name isn't unique so if you have /hello-world/ which is both a post type "post" as well as custom post type "mycpt" both will be in the loop. Even though is_single() is true and single.php is used. Which might be ok. But if you move the last created one to the trash can you will get a 404 page until the trash has been emptied.

Possible solution: Use wp_unique_post_slug_is_bad_flat_slug to make sure post type "post" and custom post type "mycpt" can't use the same post names.

Another issue which is related to the one above.

If you have a post type page, a post type post and a post type "mycpt", all three with the slug /hello-world/, the post type page will be shown. If the page is moved to the trash a 404 will be displayed. If the trash is emptied it will show as my previous comment.