Action Adventure: Zone of the Enders 2: the Second Runner
RPG: Xenogears
Racing: Mario Cart 64
Fighting: Super Smash. Bros.
Sport: Mario Tennis
FPS: Red Faction II
Puzzle: Tetris
RTS: Starcraft
SRPG: Final Fantasy: Tactics or Makai Senki: Disgaea

Note: I really don't play a lot of games on many of these genres. Thus the game choice is kind of suspect.