The best platformer games like Donkey Kong Country are Donkey Kong Country 2, Donkey Kong Country: Tropical Freeze and Super Mario World. Below are 9, each with the reason it is here — and, at the end, what filtering to platformer costs you.
Filtering to platformer means you want the movement itself to be the pleasure. It is the right filter when the thing you liked was getting from one place to another.
What you get: A game you can judge in ten seconds. Platformers live or die on the jump, and a good one feels right before anything else happens.
What made Donkey Kong Country worth playing in the first place is covered on the full list. This page is the platformer part of that answer, and only the platformer part.
The best of the trilogy, and much harder.
The modern entry, and one of the best platformers ever made.
The SNES platformer it competed with, and the secret exits are the same idea.
The most polished 2D platformer since, with the same secret-hunting.
The other great SNES platformer, and far stranger.
If the precision was the appeal more than the collecting.
Made deliberately to be Donkey Kong Country again.
Beautiful, precise 2D movement with more exploration.
The same feel translated into three dimensions.
Weight. Platformers rarely carry a story you will think about afterwards, and the collectible-led ones can become checklists if you let them.
The full list is on games like Donkey Kong Country — 9 picks, no genre filter.
Donkey Kong Country 2. The best of the trilogy, and much harder.
No. Donkey Kong Country released on the Super Nintendo, but a recommendation is only useful if you can play it, so the picks above span platforms and eras. What they share is the genre and the reason.
Whether the move set is complete early. The great ones give you everything in the first level and spend the rest asking better questions of it; the weak ones drip-feed abilities to disguise thin level design.
Different games. 2D platformers are about precision on a single screen; 3D ones are about reading space and camera. Each pick below says which it is.
By hand. Each pick was written for the full Donkey Kong Country list first, on what the game actually does, and this page is the part of that list that is platformer — never a genre tag lookup, which is how "best racing games like Super Mario 64" used to be a page on this site.
Or read the platformer guide for where to start in the genre generally.