Hmm, I like lots of games from lots of genres, so this is a tough one...
Amiga- Any type of shooter (Chaos Engine, R-Type, Sidewinder), followed by RPGs + beat-em-ups in a tie (Black Crypt + IK, Barbarian), then flight sims (Knights of the Sky, Birds of Prey, Battlehawks.)
PC: RPGs like Baldur's Gate, but also worthy FPS like Half Life and Thief 2
Arcade: Easy! Beat-em-ups all the way! Whether they are scrolling BEUs (Final Fight) or tournament BEUs (Street Fighter), they all rock.