Wow is this thread going to be subjective. But who cares? Besides, I'm thinking pretty much no one is going to be an authority on what the "best" game in every one of those genres is. As such, I'm going to just list my favorites in each.
Let's do this:
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FPS: Half-Life series - most notably Episode 2
RPG: Toss up between Earthbound and Diablo 2 (yes the latter counts)
MMORPG: Earth and Beyond - only MMO I ever really enjoyed, until EA pulled a dick move and shut it and Westwood studios down.
Strategy: Toss up between Homeworld, Sins of a Solar Empire, and Populous the Beginning
Horror: 'Nother toss up between Half-Life 1 and Eternal Darkness
Music: Oh my....tough one. But uh...anything by Rare (pre-MS), Valve, most (but not all) of IDs game music, Paul Ruskay, Jeehun Hwang, and Jesper Kyd. Unless you mean best music "game", then...um....Audiosurf?
Sports: Tecmo Bowl - hands down
Puzzle: Tie between Portal 2 and Lode Runner: Returns (PC)
Simulation: Either MechWarrior series or Freespace series
Action: Three way tie between Banjo Kazooie, Conker's Bad Fur Day, and Donkey Kong Country (all by Rare, coincidentally)
Fighter: Super Smash Bros - any really, but most notably the first
Adventure: Myst or Zork
Best Story: Ooph. 'Nother toughy. Going to have to say either Homeworld or Half-Life
Best Franchise: Super Mario Bros - there are some I like more, but the Mario franchise offers me so much variety; even if it is rehashing itself a lot lately.
GENRES NOT LISTED FOR SOME ODD REASON:
Racing: Wipeout series or Mario Kart
Shmup: Ikaruga
Third person shooter: Honestly? Spacemarine. And I'm not even a fan of Warhammer 40K.
Stealth: Either Thief (series) or Splinter Cell (series)
Action RTS: Dota 2 - maybe a tad too "new", but I'm loving the hell out of it.
Sandbox: Kinda hard to pick between Garry's Mod and Terraria, but I think I'll have to go with Gmod. Game's a thousand ways amazing.