Not that it matters as it appears no one here got it as their first game.

In the order that I had the systems...

Commodore 64: Gorf
Atari 7800: Pole Position
NES: Adventure Island
GameBoy: Tetris
SNES: Actraiser
VirtualBoy: Panic Bomber
N64: Mario 64
Genesis: Ecco the Dolphin
Sega CD: Snatcher
Saturn: NiGHTS
Playstation: Final Fantasy VII
NeoGeo Pocket Color: Metal Slug 1st Mission
Dreamcast: Sonic Adventure (importing a Japanese system at the end of November '98 was probably the best gaming decision I ever made)
PS3: Grand Theft Auto IV

NES - Super Mario Bros.
Genesis - Sonic
TurboGrafx-16 - Keith Courage In Alpha Zones
SegaCD - Sewer Shark, and I think Sonic CD is the first game I bought
32X - DOOM
SNES - Super Mario World
Saturn - Resident Evil
PSX - Resident Evil
N64 - Mario 64
Dreamcast - Soul Calibur
PS2 - Resident Evil CODE:Veronica X
Gamecube - Resident Evil
XBOX - Halo
360 - Gears of War
Game Boy - Tetris
Lynx - California Games
Game Gear - World Series Baseball
GBA - Castlevania: Circle of the Moon
PSP - Untold Legends
DS - Resident Evil Deadly Silence

I still have all systems and games except for the Lynx and Game Gear, which I wish I never got rid of.
If it isn't obvious, Resident Evil is a system seller for me. Wherever the series goes, I go.

NES - Super Mario Bros. / Duck Hunt
Genesis - Sonic
N64 - Starfox
Dreamcast - Jet Grind Radio
PS2 - Grand Theft Auto 3
360 - Tomb Raider Legend
PS3 - Heavenly Sword
Game Gear - Sonic
DS - New Super Mario Bros.

That's all.

NES - Tiger Heli
Sega Master System - Ghostbusters
Turbo Grafx 16x - Bonk
Genesis - Road Rash/Skitchin'
SNES - Zelda: A Link to the Past
Gameboy - Kirby's Dreamland
Gameboy Color - Pokemon Silver / Zelda: Link's Awakening DX
Playstation - Monster Rancher 2
Dreamcast - Crazy Taxi
PS2 - Orphen
XBOX - Halo
Gamecube - Zelda: Four Swords
Gameboy Advance - Pokemon Ruby
Nintendo DS - New Super Mario Bros.
Wii - Zelda: Twilight Princess
XBOX 360 - Guitar Hero II
Playstation 3 - MGS4

...and i'm spent.

Atari Lynx - Gauntlet, Gates of Zendacon, Electrocop and California Games (a packin)
Megadrive - Mickey Mouse and the Castle of Illusion
N64 - Zelda (but played the hell of a borrowed N64 earlier)
PS1 - Gran Turismo and Wild Arms (but same as the N64, had easy access to one earlier and played the crap out of FF7)
PS2 - Silent Hill 2 and FFX
Xbox - Halo and JSRF packin discs (but housemate had an Xbox at launch with Halo)
GC - Mario Kart DD
DS - Nintendogs
PSP - Lumines 2
360 - Bioshock, Mass Effect and Halo 3

Atari 2600 - Breakout
Atari 5200 - Defender and Pacman
NES - Can't remember some soccer game and a baseball game
Sega Genesis - NBA Jam
PS1 - Battle Arena Toshinden
N64 - Mario 64
Dreamcast - Crazy Taxi and Jet Grind Radio
PS2 - Bouncer and Tekken ?
PS3 - Motorstorm, Assassin's Creed and Oblivion

NES - Super Mario Brose/Duck Hunt
Gameboy - Tetris
SNES - Super Mario World
Game Gear - Sonic
Virtual Boy - Baseball
N64 - Mario 64
PS1 - Tomb Raider
Gamecube - Luigi's Mansion
PS2 - Grand Theft Auto 3
Gameboy Advance - Super Mario Bros. 2
Xbox - Knights of the Old Republic
DS - Meteos
Wii - Wii Sports
Xbox 360 - Gears of War

Nintendo Game Boy
- Tetris (came with it)

many years later ...

PlayStation 3
- Resident Evil 4 (PS2 game; the first thing I bought to play on the PS3)
- Unreal Tournament III (first PS3 game)

Game Boy: Tetris
Genesis: Sonic the Hedgehog
PlayStation: Spyro - Ripto's Rage
Game Boy Advance: Pokémon Sapphire
Xbox: Oddworld - Munch's Oddysey
GameCube: Super Mario Sunshine
Xbox 360: Kameo
Nintendo DS: Castlevania - Portrait of Ruin

Sega Master System: Hang On/Safari Hunt (pack in w/ system)-Grand Prix was 1st separate game
NES: Super Mario Bros. (pack in)-Double Dribble was 1st separate game
Gameboy: Tetris (pack in)- Baseball was 1st separate game
Sega Genesis: Altered Beast (pack in)- Tommy Lasorda Baseball was 1st separate game
Turbo Express (portable Turbografx-16): Vigilante
SNES: Super Mario World (pack in)- UN Squadron was 1st separate game
Playstation: WWF Wrestlemania: Arcade Game
Nintendo 64: Wave Race
Dreamcast: NFL 2K
Playstation 2: Madden 2001
Game Boy Advance: Tony Hawk
Xbox: Halo (pack in)- Max Payne was 1st separate game
Xbox 360: Madden 2006
PSP: Grand Theft Auto: Liberty City Stories
Playstation 3: MLB '07: The Show

Off the top of my head, not including pack-ins, and in the order I bought them...

c64: Root Beer Tapper!
Atari 2600: Packman
NES: Megaman II
Genesis: NHL Hockey
SNES: SFII Turbo and FFIII
Gameboy (pocket): Tetris!
PS1: NFL Gameday, Ridge Racer, and Tekken
Sega Saturn: Darkstalkers
Neo Geo Pocket: Samurai Shodown II
Dreamcast (launch day 9.9.99!): Soul Calibur and Ready2Rumble
XBOX (launch day): PGR and THPS2x
GBA (launch day): Super DodgeBall Advance
360 (9.9.06...the 7th anniversary of the DC!): NHL '07 and Burnout Revenge
PS2: Bombastic and Midway Arcade Treasures 2 (this was Jan '07)

The order in which I had them:

C64 - A whole bunch of games if I remember correctly. Omega Race, Centipede, Pac-Man, Defender ...
NES - SMB/Dhunt pack in. MT's Punch Out, Super Mario 2, Top Gun, Zelda ....
Gameboy - Tetris
SNES - Super Mario World and F-Zero
Genesis - Sonic 1 & 2
Game Gear - Some baseball game
Sega CD - Sewer Shark, WWF Rage in the Cage, SW:Rebel Assault
3D0 - Alone in the Dark and Super Street Fighter 2 Turbo
N64 - SM64, Wave-Race
Sega Saturn - X-men v.s. Street Fighter (Import)
XBox - KOTOR, Halo
XBox 360 - Call of Duty 2, Kameo, Need for Speed, King Kong, Madden 06 ....
PS3 - Metal Gear Solid 4, Silent Hill 5
Wii - I only have Wii Sports and the RE Gamecube games!
PS2 - Silent Hill: Orgins

NES: Super Mario Bros./Duck Hunt/World Class Track Meet

Gameboy: Super RC Pro-Am

SNES: Super Mario World

N64: NFL Blitz

Playstation: Metal Gear Solid

PS2: Metal Gear Solid 2

Gamecube: Legend of Zelda: Wind Waker

GBA: Mario Kart Super Circuit

Xbox: Halo

360: Oblivion

DS: New Super Mario Bros.

Wii: The Legend of Zelda: Twilight Princess

PS3: flOw

Atari 2600: Basketball or Adventure
NES: DK3
Gameboy: Never purchased/given a game.
SNES: SMW
Sega Genesis: R-Type
NEC Turbo Grafx 16: Never purchased/given a game.
PSP: SOCOM:FTB
PS2: GH1
Xbox 360: COD4

GameBoy: Star Trek: 25th Anniversary, Star Wars
(never had a NES, SNES or Genesis; played computer games for years)
N64: Diddy Kong Racing
PSX: WWF Attitude
Dreamcast: Ultimate Fighting Championship, Rainbow Six
PS2: WWF Smackdown: Just Bring It, Tony Hawk 3
XBox: Chronicles of Riddick
XBox 360: Rainbow Six Vegas

Since most systems used to come with a game, I have listed both the pack-ins (if applicable), and the first separate purchases for each system I have owned...these are all in order in which they were purchased as well:

Atari VCS (before it was rebranded as the 2600!): Combat (pack-in), Pac-Man (separate)
NES: Super Mario Bros (pack-in), Donkey Kong (separate)
Sega Master System: Hang-On (pack-in), Double Dragon (separate)
Gameboy: Tetris (pack-in), Super Mario Land (separate)
Sega Genesis: Altered Beast (pack-in), Ghouls N Ghosts (separate)
Super NES: Super Mario World (pack-in), Street Fighter 2 (separate)
Sega CD: Sewer Shark (pack-in), Sonic CD (separate)
Sega 32X: Tempo
Sega Saturn: Clockwork Knight
Sony Playstation: Battle Arena Toshinden
Nintendo 64: Super Mario 64
Game Boy Color: Legend Of Zelda: Link's Awakening DX
Sony Playstation 2: Madden NFL 2001 (had to get it as part of an Amazon launch bundle)
Nintendo Gamecube: Star Wars Rogue Squadron II: Rogue Leader
Game Boy Advance SP: Super Mario Advance 3: Yoshi's Island
Xbox: Jet Set Radio Future/Sega GT 2002 (pack-in), Buffy The Vampire Slayer (separate)
Nintendo DS: Super Mario 64 DS
Nintendo Wii: Wii Sports (pack-in), Legend Of Zelda: Twilight Princess (separate)
Sony Playstation 3: Guitar Hero III

SMS-Alex Kidd in Miracle World
NES- SMB/Duck Hunt
SNES-Super Mario World
Genesis-Sonic
Game Boy-Tetris
Lynx-Warbirds
PSX-Resident Evil
Xbox-NCAA Football/Top Spin pack in.
DS Lite-Brain Age
Xbox 360-Gears of War

I've only ever owned a xbox 360 and the first game I purchased was GH III and this was back in like August.

This is a great thread BTW.

NES: Super Mario Bros. 2 (we waited and Super Mario Bros/Duck Hunt was no longer included)
SNES: Gosh, can't remember besides Super Mario World (packed in)
PS1: I think NFL Gameday
N64: WCW/NWO Revenge
PS2: Gran Turismo 3
Gamecube: Rogue Squadron 2/Luigi's Mansion
X360: COD2
Wii: Warioware Smooth Moves
DS: Super Mario 64 (a bundle) followed by Warioware Twisted
GBA: Mario Kart Super Circuit

Sega Genesis-Sonic 1
N64-WCW/nWo Revenge
PS1-Jet Moto&NFL Blitz
PS2-NHL 2001

NES - SMB/Duck Hunt (pack in), Contra
SNES - SMW (pack in), Pilotwings
N64 - Legend of Zelda: Ocarina of Time
Gamecube - Super Smash Brothers
Wii - Wii Sports (pack in), Legend of Zelda: Twilight Princess
Playstation - Tomb Raider
PS2 - GTA III
PS3 - Ratchet & Clank: Future Tools of Destruction
360 - Gears of War/Dead Rising (bought at the same time)
DS - New Super Mario Bros.
PSP - Star Wars Battlefront 2 (pack in), Ratchet & Clank: Size Matters

NES - Super Mario Bros / Duck Hunt (Pack-in)
Gameboy - Tetris (Pack-in)
SNES (Didn't own, but first played) - Super Mario World
Sega Genesis - Sonic the Hedgehog (Pack-in)
Sega CD - Sewer Shark (Pack-in)
Sega 32X - Never actually owned a game for it / Doom (first played)
Nintendo 64 - Super Mario 64 / Wave Race
Sega Dreamcast - Psychic Force 2012 (came with it) and Sonic Adventure
Playstation - Tomb Raider 2 (My parents bought me the console, and made me buy the game from them. It makes sense in a way but it's still funny to me)
Playstation 2 - I think it was SSX.
Xbox - Halo
Xbox 360 - Perfect Dark Zero (uggh)
Wii - Wii Sports
Nintendo DS Lite - Brain Age (came with it)
Playstation 3 - Uncharted

*Commodore 64 - Mickey's Space Adventure (A Disney adventure game developed by Sierra.)
*PC - The Incredible Machine / Doom 2

My god, all that money wasted allocated on gaming

Hmm... Tricky. Not counting pack-in games

Atari 2600 - Robot Tank

Coleco Vision - MouseTrap

NES: Teenage Mutant Ninja Turtles

SNES - I want to say it was called "Super High Impact Football." Does that sound right? I'll look it up in a bit.

Playstation: WWF Warzone

Playstation 2: Some iteration of Virtua Fighter... Maybe 2?

GameCube: Super Mario Sunshine

GBA: Wow... I honestly don't remember. I bought it solely for the connectivity with Animal crossing. I think it was a badly ported Medal of Honor game.

Xbox: Fable. Only game I ever owned for the system.

DS: Mario Kart came as a packin with my system, and Animal Crossing: WW

PSP: None... won it. Rented Lumines and Ridge Racer... it sat for a year, and then I sold it to buy my 360.

360: Madden 07

Wii: Twilight Princess.

PS3 is coming soon, and I really want to play Uncharted.

Atari 2600: Pac-Man
NES: Super Mario Bros.
GameBoy: Super Mario Land
SNES: Final Fantasy II(IV) - Super Mario World came with my SNES but FF II was the first game I purchased.
Turbo Grafx 16/Express: Bonk's Adventure
N64: Super Mario 64
Playstation: Ridge Racer
GameBoy Advance: Castlevania: Circle of of the Moon
Sega Dreamcast: Soul Calibur
Playstation 2: Ridge Racer V
GameCube: Star Wars Rogue Squadron II: Rogue Leader
Xbox: Star Wars: Knights of the Old Republic
PSP: Ridge Racer
Nintendo DS: Super Mario 64 DS
Playstation 3: Ridge Racer VII
Xbox 360: Gears of War
Wii: Zelda: Twilight Princess