Listing the first version I played in most cases, but man, it is really tough to narrow some of these to just one game and not list an alternate or two. Not surprisingly, RPGs take center stage.
Dreamcast: Shenmue
GameCube: Resident Evil 4
Neo-Geo: King of Fighters 2002
Nintendo: Dragon Warrior
PC: Star Wars: Knights of the Old Republic II
PlayStation: Metal Gear Solid
PlayStation 2: Shin Megami Tensei: Persona 3
PlayStation 3: Valkyria Chronicles
Saturn: Panzer Dragoon Saga
Super Nintendo: Final Fantasy III
Xbox: Fatal Frame II: Crimson Butterfly
Xbox 360: Mass Effect 2
Opted to stick to platforms I'm pretty intimately familiar with, and I'm so backlogged on 3DS, PSP, DS, Vita, and Wii games, I don't think I could fairly pick anything.