Only gonna attempt to recommend one game per platform to save yuge lists. No ports either of course.
NES: Super Mario Bros 3 (best 2D platformer of all time).
SNES: Donkey Kong Country 2 (this one isn't far behind. Starts off feeling all gay and racist
lol
monkeys on a pirate ship (I'm fucking around, not racist at all) but gets a lot better).
Genesis: ehhh...lots of good games but nothing I'd consider must-play. I loved this system as a kid and the games are great but there is nothing that will blow you away. If you're really into retro regardless then I'll go with Streets of Rage 1 (beat em up).
N64: Toss up between Turok1/2, Doom 64 (FPS) or Duke Nukem: Zero Hour (TPS). Each are equally gud. Ignore the entire rest of the library. Collectathons galore, inferior versions of PC or PSX games, and not a lot else.
PSX: Castlevania: Symphony of the Night (the OG metroidvania ARPG) or Final Fantasy VII (I dgaf, love this game and you should too).
MS-DOS: Doom or Quake (FPS. but of course!)
Windows 95/98/XP: Deus Ex or System Shock 2 (ARPG. but of course!)
Gameboy: James Bond 007 (for real. Really cool action-adventure game).
Gameboy Color: Legend of Zelda: Link's Awakening DX (solid action-adventure, made 98% of the gameboy library look like total shit by comparison).
Gameboy Advance: Golden Sun 2 (JRPG) or Advance Wars (fast paced Turn-based Strategy)
Nintendo DS (may as well throw this in since the majority of the games are retro in design): Castlevania: PoR or OoE.
Played the crap out of all the above systems. Didn't play much Dreamcast, Amiga, Commodore 64, Sega Saturn etc to have a valid answer. A little of each.