There’s plenty of fun platformers, puzzlers, action games, and arcade ports on portable systems like the Game Boy and Nintendo DS etc. Nintendo has always been a master of “pick and up and play”-ability.

Some classics include the Super Mario Land and Wario Land series, Kirby, Donkey Kong GB (it’s the original arcade game plus a brilliant puzzle platformer), and the Zelda titles. There’s even some lighter weight RPGs like the Pokémon games etc.

There’s also Game Boy ports of the Game & Watch titles which are great short time wasters - Game Boy Gallery and Game & Watch Gallery series.

• SNES

  • the Donkey Kong Country trilogy, just good platformers; first one hasn't aged as well as people think, if you don't enjoy it skip ahead to 2 and come back later if you feel like it
  • The Lost Vikings 1 and 2, fantastic character-switching co-operative (but playable solo) level-based puzzle-platformer; don't play the sequel on newer platforms like PlayStation, they tried to modernize the graphics and only succeeded in making them hideous, stick with SNES
  • Star Fox 2, as long as you enjoy dated 3D and poor framerates, this is the most ambitious 3D game on any 16-bit console, with a very replayable campaign full of hidden unlockables that differs on each play
  • Super Mario All-Stars & Super Mario World, just an enhanced compilation of 4-5 seminal 2D platformers; while you could use save states on the NES versions, the SNES versions all support native saves, so easier to pick up and play
  • Tetris Attack, nothing to do with Tetris, just a match-three puzzler with deep, engrossing mechanics that can keep you interested long-term; also Panel de Pon with a translation hack if you want more
  • Top Gear, fantastic competitive racer with the line-scroll road effect you know from classic arcade games like OutRun and a killer soundtrack; pick the white car
  • Yoshi's Island, another classic platformer

• Mega Drive/Genesis

  • Gauntlet IV, better than the arcade original, this is M2 (now known for developing emulators for many classic systems) flexing with some RPG mechanics added to the traditional Gauntlet gameplay
  • The Lost Vikings, SNES version is better but there's a few brand new levels in this one if you want more
  • Sonic the Hedgehog 3 & Knuckles, basically the first Sonic that noticed it's a home game, so it has saving and improved replayability with multiple characters, paths and unlockables

• Game Boy

  • Donkey Kong, this is not a port of arcade Donkey Kong, it's a full-blown puzzle platformer you can play one level at a time
  • Kirby's Dream Land 2, easy-to-finish platformer but with tons of content if you're playing "properly", using the sort of rock-paper-scissors logic to use the right powers to enter the secret areas
  • Super Mario Land 2: 6 Golden Coins, just a proper "mid-period" (SNES-esque) Mario platformer which you can play over as long as you like

• Game Boy Color

  • Balloon Fight GB, predates Flappy Bird by literal decades but like if that was a proper game with a campaign
  • The Mummy, based on the Brendan Fraser movie and an awesome Konami puzzle-platformer with short individual levels and password save (use save states on emulator), super underrated
  • Pokémon Puzzle Challenge, Tetris Attack but with Pokémon if you prefer that theming
  • Wendy: Every Witch Way, based on some kind of comic book I think, developed by WayForward who developed the Shantae games and then branched off into Yacht Club Games (Shovel Knight), Wendy is a gravity-flipping platformer where you're in control of which direction is up or down

• Game Boy Advance

  • Advance Wars 1 & 2, adorable turn-based strategy war game with a campaign based around small, self-contained levels, except for a few huge ones
  • Drill Dozer, developed by Game Freak (Pokémon), is a sort of level-based Metroidvania platformer with lots of backtracking to older levels as you unlock new abilities
  • F-Zero: Maximum Velocity, the only true sequel to SNES F-Zero, don't at me
  • Game Boy Advance Video: Shrek, endlessly replayable
  • Metroid: Fusion and Zero Mission, they're Metroid games
  • Super Mario Advance 4: Super Mario Bros. 3 e-Reader version, I think you can get this officially on Switch somehow, but you've been to the Internet before; this officially-released modified version of the original game includes a bunch of brand new levels previously distributed only on scannable cards, I'm not telling the whole story but they remix elements from the first four mainline Mario games into basically a whole original game. This is New Super Mario Bros. this is Super Mario Bros. 5, still don't at me

Secret of Mana SNES is special if you've never played it.

https://gamefaqs.gamespot.com/psp/934454-jeanne-darc?validate=1 is really good and chilled to play. It's a SRPG that doesn't have a huge learning curve but is still satisfying.

Puzzle Quest
Bejeweled
Peggle
Plants vs Zombies
Punch Out
Pokemon Mystery Dungeon

• Puzzle Games
  • Super Puzzle Fighter 2 Turbo (GBA, other platforms)
  • Puzzle League (GBA, paired with Dr Mario same cart)
• Roguelikes
  • Dragon Crystal (Sega Master System, GG too)
  • Shiren the Wanderer (Nintendo DS, others too)

I find puzzle & roguelikes the best for just diving in for a little bit and not worrying about when you drop off.

Pokemon Pinball