Ill assume this just pertains to PS2:
Action--GTA3. Since i dont have Metal Gear 2 yet.
Adventure--Devil May Cry, its fast paced and bad @ss. I dont recommend this game as a buy though. It\'s too short, and too repetitive to play over again. It should be a long rent.
Fighting--DOA2...she kicks high, and mainly there arent really that many other good ones out there.
Platform--Jax and Dexter. This game is beautiful--its like the Crash and Banjo and Kazooie, but its beautiful.
Puzzle--Tie between Silent Hill 2 and Ico.
RPG--I would still go with an PSX game like Vagrant Story
Racing--GT3
Sports--Tony Hawk 3. Its like 2 with all the same stuff, just more of it, my friends and i play this for hours