PS2: Wave Rally,Gt3, Jak & Daxter 

Ps3: Oblivion GOTY

PS: Street Fighter Collection

Game boy: Super Mario Land

Game boy  Advance:  Final Fight

Gamecube: Medal of Honor Frontline

Xbox: Halo 2

Nintendo DS: Nintendogs

Nintendo Wii: Wii Sports, Resident Evil 4

Xbox 360: Gears of war, UFC 2009 Undisputed

Let's do this!

Game Boy: Tetris

Mega Drive (Genesis): Sonic the Hedgehog

Saturn: Daytona USA

Nintendo 64: (console and all the games I have for it were offered to me at the same time)

PlayStation: Silent Hill

PlayStation 2: Onimusha: Warlords

Gamecube: Luigi's Mansion

Dreamcast: Sonic Adventure

Game Boy Advance: Castlevania: Circle of the Moon

NDS: New Super Marios Bros.

PSP: Grand Theft Auto: Liberty City Stories

PlayStation 3: Assassin's Creed (oh lord...)

Nes: super mario bros

GB: mario world

PSX: Final fantasy VII

PS2: GTA III

PSP: God Of War: Chains Of Olympus

PS3: GT5

SNES: Super Mario World
GB: Tetris and Killer Instinct
N64: Super Mario 64 and Pilotwings 64
GC: Mario Kart Double Dash!!

Lol I can't remember my first game from anything but this gen

wii - wii sports (surprising right :D)

ps3 - LBP

PSP - Metal Gear Solid: Peace Walker

DS - New super mario bros

PS1 - NHL 98

PS2 - NHL 2000

PS3 - Madden 07

360 - Call of Duty 4

PSP - NFS Underground Rivals

GB - I think Pokemon Red

GBA - I think Super Mario Bros 3

3DO - Road & Track Presents The Need For Speed

360 Slim - Halo Reach

NES: Super Mario Bros/Duck Hunt of course! Came with the console. Still own it to this day, and it's still just as fun. My gun no longer works though :(

SNES: Super Mario World/ Final Fight. The former obviously came with the console, got the second for christmas along with it.

N64: Super Mario 64. Was the only N64 game I owned for awhile, as the games/console were expensive. But I was so wowed by the 3D graphics it didn't matter.

Gamecube: Luigi's Mansion, Tony Hawk 3

Dreamcast: Space Channel 5, my sister was obsessed with this game, so she got a DC just for it. I bought Crazy Taxi soon after.

Xbox 360: Kameo. Fun, underrated platformer, although a little too short. I was excited to check out Rare's latest offering so I got the console with this game.

Wii: Zelda TP, Red Steel, Super Monkey Ball, Wii Sports

DS: Super Mario 64 DS, some Sims game..

Well, I go back:

Atari 2600: Combat!
Atari 800: Star Raiders
PC: The Original Heros (later became the Might and Magic Series)
upgraded PC: Sims II
Wii: Twilight Princess, Super Mario Galaxy and Wii Fit - it was an expensive first day.
Xbox 360: Halo:Reach, Pinball FX and Limbo.

X360 Kinect: Kinect Adventures and Your Shape: Fitness Evolved.

And if I had a PS3 I would get Flower and God of War III.

Gamecube - Wave Race: Blue Storm, Star Wars Rogue Squadron II (launch, I think)
Xbox - The Elder Scrolls: Morrowind, Pro Evolution Soccer 4
DS - Polarium, Zoo Keeper, Wario Ware Touched (launch)
Wii - The Legend of Zelda: Twilight Princess (launch)
PSP - Lumines, LocoRoco, Patapon
PS3 - Flower, Uncharted, Valkyria Chronicles
360 - Mass Effect, Rez HD

PC - Indiana Jones and the Fate of Atlantis
(386 PC/MS-DOS; I know PC doesn't make sense at all in this thread, but I thougt I'll list the first videogame I ever bought for any system anyway;  IMO Indy&tFoA is still the best game based on a movie franchise ever created)