This shall be the toughest thing I will ever do :'(
NES: Legend of Zelda
SNES: Super Metroid
N64: Legend of Zelda Ocarina of time
Gamecube: Legend of Zelda Wind Waker
Wii: Legend of Zelda Skyward Sword

Sega Genesis: Sonic CD
Sega Saturn: Virtue Fighter 2
Sega Dreamcast: Sonic Adventure

PS1: Final Fantasy VII
PS2: MGS 3
PS3: MGS 4

Gameboy: Legend of Zelda: Links Awakening
GB Color: Legend of Zelda: Oracle of Seasons
GB Advance: Pokemon Ruby Version
DS: Mario and Luigi Bowsers Inside Story
3DS: Kid Icarus: Uprising

PSP: Monster Hunter Freedom Unite
PSV: Gravity Rush

Xbox: Halo
360: Halo Reach (soon to be Halo 4)

NES: Castlevania
Gameboy: Castlevania 2 Belmont's Revenge
Sega Genesis: Phantasy Star 4
Sega CD: Dark Wizard
Super NES: Final Fantasy 6
Nintendo 64: Nightmare Creatures
PS1: Brigandine Legend of Forsena
GBA: Castlevania Circle of the Moon
Dreamcast: Phantasy Star Online.
PS2: Wild ARMs 3
GameCube: Phantasy Star Online Episode 1 and 2
Wii: Super Smash Bros Brawl
DS: Retro Game Challenge
PSP:  Castlevania Dracula X Chronicles
PS3: Grand Theft Auto 4 Complete
3DS: I own over 10 games and still not satisfied giving an answer because none of the games are that good. But if I must. Mario Kart 7.
Vita: Little Big Planet(only because I don't own Gravity Rush and haven't played more than the demo, so this may change.)

GB: Pokemon Red/Pokemon Blue (I fixed the rest, but it's fair to have two games here, dontcha think?)
GBC: Dragon Warrior III
GBA: Pokemon Mystery Dungeon: Red Rescue Team
DS: Dragon Quest IX
3DS: New Super Mario Bros 2
Wii: New Super Mario Bros Wii

EDIT: Not that I've been much of a PC gamer in the last ~10 years, but...

PC: Age of Mythology: The Titans

I can't take OP seriously when they have Skyrim listed under favorite game on PS3.

Still a good questions so by memory here it goes

Atari: don't remember

Coleco Vision: don't remember

Commodore 64: Some football/soccer manager type game

PC: Quest for Glory, Wing Commander, and Age of Mythology

NES, SNES, N64: Zelda

GB: Secret of Mana

GBA, DS: Fire Emblem

PSOne: Never owned

PS2: Ico (ported to PS3)

PSP: God of War (ported to PS3)

Gamecube: Super Smash Bros Melee

Wii: Zelda

PS3: Uncharted

XBox: Ninja Gaiden Black and Dead or Alive Ultimate

X360: Halo

PS-Vita: Rayman Origins

NES: Nintendo World cup, SMBros.
SNES: SuperMario World, DKC
N64: Zelda Majoras mask, Perfect dark
GC: Resident evil 0, Mario Sunshine
Wii: Metroid Prime trilogy, Zelda Skyward sword

MegaDrive: Sonic 2, Sensible soccer
Dreamcast: Soul calibur, Shenmue 2

PSX: Resident evil Directors cut, GT
PS2: GT3, DBZ Budokai 3

GB: Donkey kong land
GameGear: Sonic Triple trouble
GBA: Golden sun
DS: Suikoden tierkreis
3DS: Mario tennis

PSP: Silent hill origins

Playstation: Crash Bandicoot
Playstation 3: Vanquish

Gameboy: Pokemon Blue
Gameboy Advance: Metroid Fusion
Nintendo DS: Advance Wars Dual Strike
Gamecube: Metroid Prime

Xbox 360: Halo 3

PC: Warcraft III

Hoo boy, might have to think about a few of these...

N64: Super Mario 64 and Paper Mario (I know you said only one, but these are two of my all-time favorite games... Couldn't bear to leave either of them out!)
Gameboy Color: Pokemon Gold
Gameboy Advance SP: Pokemon Ruby
Gamecube: Paper Mario: The Thousand Year Door (though Tales of Symphonia runs it a close second)
Xbox: Fable (Say what you will about its sequels, but I had a ton of fun with this one)
Playstation 2: Kingdom Hearts! (Love the second game, too)
DS: The World Ends with You and Chrono Trigger DS (Again, two of my all-time favorites, and I'd never played Chrono Trigger before the port)
Wii: Xenoblade Chronicles (Haven't quite finished it yet, but I'm really close... Won't be able to play it for another 8 months though -_-;)
Xbox 360: Enslaved: Odyssey to the West (More people need to play this!!) and Lost Odyssey are tied for me.

Edit: Forgot about the PSP, 3DS, and PC.

PSP: Kingdom Hearts: Birth by Sleep was my favorite there.

3DS: So far, Legend of Zelda: Ocarina of Time

PC: Guild Wars!!

Nintendo Entertainment System- Super Mario Bros.
Game Boy- Tetris (I guess)
Super Nintendo Entertainment System- Super Mario World/ Donkey Kong Country
Game Boy Advance- F-zero GX
Nintendo 64- Mario Kart 64/ Perfect Dark
Gamecube- Mario Kart Double Dash
Wii- New Super Mario Bros. Wii/ Madworld

Sega Saturn- Sega Rally
Dreamcast- Chu Chu Rocket

Playstation- Gran Turismo
Playstation 2- Grand Theft Auto: San Andreas
Playstation Portable: Grand Theft Auto: Liberty City Stories
Playstation 3: Red Dead Redemption

PC: Minecraft

Game Boy: Kirby's Dreamland
Game Boy Colour: Pokemon Yellow
Game Boy Advance: Wario Ware
DS: Pheonix Wright: Ace Attorney
3DS: Kid Icarus Uprising

SNES: Yoshi's Island
Wii: Super Mario Galaxy

PSP: LittleBigPlanet psp
Vita: LittleBigPlanet Vita

PS1: Crash Bandicoot
PS2: Kingdom Hearts
PS3: LittleBigPlanet 2

Gameboy Color: Pokemon Blue

Nintendo 3DS: Super Mario 3D Land

Nintendo 64: Super Mario 64

Nintendo Gamecube: The Legend of Zelda: The Wind Waker

Nintendo Wii: Super Mario Galaxy 2

Playstation 3: Journey