Platformer - Super Mario World, I honestly just blast through this game recently and it still holds up.
Puzzle - Ico, I'm with Slothy, Ico is a puzzle game.
RPG - WoW with
this little trick
Action - Devil May Cry 3
Fighting - Street Fighter Alpha 3
MMO - Don't really play them but I'll give a shout out to FFXI.
FPS - Bioshock
Rhythm - Dance Dance Revolution
Adventure - The Legend of Zelda: A Link to the Past. If you insist on a more point and click affair, Leisure Suit Larry in the Land of the Lounge Lizards.
Racing - Mario Kart 54
Sports - Cyber Stadium Series: Base Wars
Mobile - Terra Battle or Sailor Moon Drops
Survival Horror - Silent Hill 2
4X - Civilization 2
Tactical/Strategy - Final Fantasy Tactics
Sandbox - Metal Gear Solid V: The Phantom Pain
Visual Novel - Phoenix Wright
Stealth - Metal Gear Solid 3: Snake Eater