Genesis: Sonic and Knuckles
Gameboy: Pokemon Red
Dreamcast: Skies of Arcadia
Gamecube: Paper Mario TTYD
Xbox: Psychonauts
PS2: We Love Katamari
GBA:Mario and Luigi Superstar Saga
DS: Chrono Trigger
PSP: Final Fantasy Tactics
360: Lost Odyssey
Wii: Super Mario Galaxy
PS3: Valkyria Chronicles
3DS: LoZ:OoT
PC: Portal

PS1: Final Fantasy Chronicles. Second would be Breath of Fire III
PS2: Kingdom Hearts 2. Second FFX

Wii: Super Paper Mario
GB: Pokemon Yellow (the only GB game I had pretty much)
GBA: The Legend Zelda: The Minish Cap
3DS: Kid Icarus. Second would be Tales of the Abyss (didn't play it on the PS2)
GC: Tales of Symphonia and a close second is Luigi's Mansion.
SNES: Super Mario World and Chrono Trigger
N64: Yoshi Story and Ocarina of Time

PC: Probably MapleStory because it's free. I don't have very many PC games.

edit:  Sorry I had to keep adding and changing things because I named some things wrong and forgot some consoles

SNES: Can't remember....
N64: Mario 64/Donkey Kong 64/Ocerina of Time/Mischief Makers/Banjo/Majora's Mask
GameCube: The Wind Waker/Roque Squadron
Game Boy: Some funky Spider Man Game

PS1: Can't remember...
PS2: Some Crash Bandicoot game
PS3: God of War 3

XBOX: Halo/Halo 2/Knights of the old Republic/Splinter Cell/Beyond Good & Evil
Xbox 360: Halo 3/Gears of War/GRID

PC: Age of Empires/Command & Conquer/Half-Life 2

Too difficult to choose a clear cut winner for anything except PS3.

Atari 2600: River Raid
NES: Super Mario Bros. 3
SNES: Super Mario World
Gameboy: Dr. Mario
N64: Goldeneye
Gameboy Advance: Mario Kart Super Circuit
Gamecube: Windwaker
PS2: GTA San Andreas
Xbox: Knights Of The Old Republic
PSP: F1 2005
DS: Animal Crossing
Wii: Super Smash Bros. Brawl
Xbox360: Gears of War 3
3DS: Ocarina of Time 3D
PC: Minecraft

Snes: Final Fantasy VI

PS1: Xenogears

PS2: Shin Megami Tensei: Nocturne
GameCube: Baten Kaitos
Xbox: Steel Battalion

PS3: Metal Gear Solid 4
Wii: Xenoblade
360: Lost Odyssey

GB: Pokemon Yellow

GBA: Golden Sun
DS: Radiant Historia
PSP: Metal Gear Solid: Peace Walker

That I have ever owned:

360 - Skyrim
Windows 7 Phone - Final Fantasy (since we're adding handhelds and a phone is pretty much just that)

That I have ever played:

N64 - Mario Kart 64
Xbox - Halo 2
Wii - Wii Sports Resort
PS1 - 007: Tomorrow Never Dies
PS2 - God of War
PS3 - God of War III
DS - Pokemon White Version

NES: Super Mario Bros. 3
SNES: Super Mario World
N64: Zelda OoT
GC: Super Smash Bros. Melee
Wii: Super Smash Bros. Brawl

GB: Super Mario Land
GBC: Pokemon Silver
GBA: Fire Emblem (I think it's FE6 in japan)
DS: Jump Ultimate Stars
3DS: Kid Icarus Uprising

PS1: Final Fantasy VII
PS2: Shadow of the Colossus/ Dragon Quest 8
PS3: MGS4

PSP: Dissidia Final Fantasy

Xbox: Halo 2
360: Halo 3

Dreamcast: Power Stone 2

This is difficult, some platforms I didn't buy enough games for.

N64 - Resident Evil 2
Gamecube - Metal Gear Solid: The Twin Snakes
Wii - Super Mario Galaxy

PS1 - Final Fantasy VII
PS2 - Metal Gear Solid 3
PS3 - Final Fantasy XIII

Xbox - Grand Theft Auto: San Andreas
Xbox 360 - Tales of Vesperia

Sega Mega Drive - Sonic 3 & Knuckles
Sega Saturn - NiGHTs into Dreams
Sega Dreamcast - Sonic Adventure 2

GB - Super Mario Land 2
GBC - Pokemon Silver / Gold
GBA - Pokemon Ruby / Sapphire
DS - Pokemon Heart Gold / Soul Silver (need to buy more games)

PSP - Crisis Core: Final Fantasy VII

Mega Drive: Sonic 3 + Sonic & Knucles
PS1: Metal Gear Solid
PS2: Gran Turismo 3 A-spec
GC: Super Smash Bros Melee
DS: New Super Mario Bros
X360: Forza Motorsport 3
Wii: Donkey Kong Country Returns
PS3: Demon's Souls (so far)
3DS: Mario Kart 7 (so far)

Yay, someone else who played Tomorrow Never Dies. I thought that game was great, I watched the film a few hours ago.