SNES - Mario Bros
Sega Mega Drive - Mortal Kombat 2
PS1 - FF7
PS2 - FFX
GameCube - RE Remake
PSP - Jeanne d'Arc
PS3 - Dark Souls
PS Vita - Persona 4 Golden

OK here we go.....

Atari 2600 - River Raid
NES - Super Mario Bros. 3
SNES - Super Mario World
GameboyColour - Zelda: Links awakening
N64 - Goldeneye
Gamecube - Zelda: WindWaker
PS2 - GTA: San Andreas
Xbox - Knights of the old Republic 2
GBA - Golden Sun
DSlite - Zelda: Phantom Hourglass
PSP - F1 2009
PC - Minecraft
Wii - Smash Bros. Brawl
Xbox360 - Battlefield 3
PS3 - The Last of Us
3DS / 3DSXL - Zelda: A Link Between Worlds
PSVita - LittleBigPlanet
PS4 - Battlefield 4

My List.....

Atari 2600:  River Raid

NES:   Super Mario 3

PC :   Unreal Tournament

PS2: (2 of them)  God of War

Gamecube:  Super Smash Bros.

PSP:  Pursuit Force

PS3: (2)  God of War 3

Xbox 360:   Halo Wars

Wii: Wii Sports

Started buying older systems that I can afford now, but couldn't as a kid

SNES:  Super Mario World

Genesis:  Sonic the Hedgehog

N64:  Mario Kart 64

Dreamcast:  Didn't have a Favorite Game

Xbox:   Didn't have a Favorite Game

PlayStation:  Oddworld: Abes Odyssey

Playstation Go:  God of War- Chains of Olympus

DS Lite: Guitar Hero: On Tour

DSi:   New Super Mario Bros.

PS Vita:  Killzone- Mercenary

3DS:  Super Mario 3D Land

PS4:  Battlefield 4 (so far)

Have a Sony PlayStation 3D Display also.

It was really hard to pick a favorite game from each console. I might be able to pick a favorite game out of a single genre for certain consoles. 

Still have all of these systems, some are packed away however as I don't have the room in my current living arrangement to have them all out. Plan on getting the  Wii U and Xbox One at a later time, when the paycheck allows...

Atari 2600 - Pitfall

Coleco - don't remember it enough

Nes - Super Mario Bros 3

Sega Genesis - Sonic the hedgehog

SNes - Super Mario World

N64 - Zelda OOT

Playstation - FF8

Playstation 2 - SSX Tricky

Gamecube - Metroid Prime

Xbox 360 - Gears of War

Wii - Mario Galaxy

Playstation 3 - Uncharted 2

DS - Mario Kart

3DS - Super Mario 3d Land

WiiU - New Super Mario U waiting on SMB3dW

good picks!  a lot of my faves there.

SNES
N64
Game Boy Color
Dreamcast - The House of the Dead 2
PS1 - Silent Hill and Final Fantasy VIII
PS2 - Grand Theft Auto: Vice City, Silent Hill 3, and Final Fantasy X
GBA - Pokemon FireRed and LeafGreen
GameCube - Legend of Zelda: Wind Waker and Resident Evil 4
Xbox - Project Gotham Racing
DS - Pokemon Black and White
PSP - Dissidia 012
Xbox 360 - Gears of War, Need for Speed: Most Wanted
Wii - Legend of Zelda: Twilight Princess
PS3 - Grand Theft Auto V

SNES, N64, and GBC have nothing listed since I only own the hardware itself... I don't remember playing any games on those systems.

Im not naming handhelds. but here is the order.

SNES - Link To The Past
N64 - Ocarina of Time
Playstation - Final Fantasy 7
PS2 - Shadow of colossus
Gamecube - Smash Bros Melee
Xbox360 - call of duty 4
Sega genesis - golden axe
NES - Zelda 2 (dont judge I really like z2)
Wii U - Mario 3D World

Favorite console being N64
least favorite being Playstation 1 or 2

I own every Nintendo, Sega, Sony, and XB xonsole (except for the original XB, and GameGear). I don't own a Saturn/Virtual Boy yet, but I plan to change that.

Favorite games?

NES - Final Fantasy
SNES - Chrono Trigger/Super Mario RPG
N64 - Mario Party 2/Mario Kart 64
GCN - Metroid Prime/Resident Evil 4
Wii - XenoBlade Chronicles

GB - Pokemon Red/Blue
GBC - Pokemon Gold/Silver
GBA - Advance Wars/Golden Sun/Fire Emblem: Sacred Stones
DS - Front Mission DS/Advance Wars: Dual Strike
3DS - Fire Emblem Awakening

PS1 - I have so much left unplayed, would be unfair to vote here (mostly played N64 during this gen, but I want to catch up on PS1/PS2)
PS2 - GTA: Vice City/Shadow of the Colossus/Gran Turismo 3 (but I have a lot of JRPG's I have yet to play)
PS3 - Uncharted 2/Gran Turismo 5

PSP - Gran Turismo (but again, so many unplayed JRPG's)

Gen - Ranger X
CD - Lunar: SSH
DC - Sonic Adventure 1 & 2 (but I have a few JRPG's yet to play)

XB1 - Halo
360 - Gears of War 2/Lost Odyssey/Forza Motorsport 3/Mass Effect 2

It's too hard to narrow down any console to one game, GBA for instance, I remember it for a lot of reasons.

I've always been a Nintendo/Sony gamer, but I guess growing up it's always been Nintendo primary/Sony secondary, and now it's vice versa. Didn't get into JRPG's until I was in my teens, and I have a lot of JRPG's to catch up on. However I'm not rushing it, I don't mind the age of SNES/PS1/PS2 era games, it's the plot that's most important. So I'm slowly cstching up on my backlog with 2-5 retro games a year.

Stationary consoles:

N64 - The Legend of Zelda Ocarina of Time
Gamecube - Super Smash bros. Melee
Wii - Super Mario Galaxy
PS3 - Metal Gear Solid 4
Wii U - Wind Waker HD (nice remake)

Portable systems:

Original Gameboy: Pokemon Yellow
Gameboy Color: Super Mario Bros. Deluxe
Gameboy Advance: The Legend of Zelda The Minish Cap
Nintendo DS: Mario Kart DS
Nintendo 3DS: Super Mario 3d Land

I still own all of my consoles and I don't plan on selling any of them in the future time

Playstation 1 - Final Fantasy VII

Xbox - Halo: Combat Evolved

Xbox 360 - Red Dead Redemption

Nintendo DS - Phoenix Wright: Trials and Tribulations

Nintendo 3DS - Phoenix Wright Dual Destinies

EDIT: Amazed at how many system do some people here own :(