The best arcade games like Donkey Kong are Donkey Kong Country, Mario Bros and Bubble Bobble. Below are 7, each with the reason it is here — and, at the end, what filtering to arcade costs you.
Filtering to arcade means you want short runs and a score, not a campaign. It is the right filter when what you liked was the one-more-go feeling.
What you get: A complete experience in twenty minutes, and a skill you keep. Arcade design is measured in the moment after you lose.
What made Donkey Kong worth playing in the first place is covered on the full list. This page is the arcade part of that answer, and only the arcade part.
What the character became, and a completely different genre.
The other arcade original, and best with a second player.
Single-screen arcade platforming at its most charming.
The other arcade game built on patterns rather than reflexes.
Where the jump stopped being a screen and became a world.
Same era, same single-screen discipline, superb two-player.
Momentum platforming with arcade scoring underneath.
Story and progression. Arcade games repeat on purpose, and if you want to be pulled somewhere they will not do it.
Specifically, these are strong answers for Donkey Kong that are not arcade games, so they are not on this page:
The full list is on games like Donkey Kong — 9 picks, no genre filter.
Donkey Kong Country. What the character became, and a completely different genre.
No. Donkey Kong released on the Nintendo Game Boy, 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.
They were designed to end. A machine that let you play for an hour on one coin did not earn its floor space, so difficulty was the business model.
Constantly, mostly by small teams. Several of the best beat-em-ups and run-and-guns of the last decade are new games built on forty-year-old design rules.
By hand. Each pick was written for the full Donkey Kong list first, on what the game actually does, and this page is the part of that list that is arcade — 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 arcade guide for where to start in the genre generally.