Nintendo 64
Gameboy Color
Gamecube
Gameboy Advance
Gameboy Advance SP
DS
Nintendo 64 (Bought for $20 after my Mom gave my first one away... along with all my games... )
Wii
NES
DS Lite
3DS
Wii U

Awesome! I have my systems listed chronologically in the order I acquired them on my Backloggery page. Here they are:

Atari 2600 (RIP, I no longer have my Atari)
NES
Gameboy
SEGA Genesis
Super Nintendo
SEGA CD
Gameboy (Blue Ice Limited Edition)
Nintendo 64
Gameboy Advance (sold this for the SP)
e-Reader
Gameboy Advance SP
Gamecube
Playstation 2
Nintendo DS Lite
X-Box (original - sold it along with all my games...ewww, microsoft)
Wii
Nintendo 3DS
Playstation 3

@berzerk99 I re-added your 3DS FC. My new one is in my signature.

Mine were:

Nintendo Famicom (you read that right)

Game Boy Color (broken by the 3-4 years old version of me, along with the Famicom)

Game Boy Advance

Nintendo DS Lite

Nintendo GameCube

Nintendo 3DS (Zelda color! >:3)

Game Boy Micro (Famicom edition >:3)

@CaptainMorpheus - I'll add you as soon as I get home and near my wifi. Also....COOL!!!

NES
N64
GBC
GBA
GCN
GBASP
DS
Wii
Xbox 360
3DS
Wii U

PC.
Playstation 2.
Gamecube.
DS Lite.
Wii.
3DS.
Wii U.

Game Boy Advance (special edition Pikachu and Pichu one)
Game Boy Advance SP (special edition Leaf Green one)
(Another) Game Boy Advance SP (my Leaf Green one broke... )
Playstation 2
DS Lite
Wii
DSi XL
3DS (sold the XL for it)
Playstation 3
Nintendo 64
Playstation Portable

The second SP and the PS2 could be the other way around, but that was so long ago I honestly can't remember.

Game Boy Color
Game Boy Advance
Nintendo DS
Nintendo DSi
Nintendo 3DS
Wii U

Systems I used to have:
Atari 2600 Jr. (My Mom gave the system to my younger brother's friend when I was growing up) - got it after the Intellivsion II

Systems I currently have:
Intellivision II
Atari 7800
Genesis
32x
Playstation
Game Boy Color
Sega Master System
Snes
Dreamcast
Playstation 2
Game Cube
Gameboy Advance
Neo Geo Pocket Color
Atari Lynx
Nintendo DS
Wii
Game Gear
Saturn
PSP
Nintendo DS XL
3DS XL

Light Bright
Nes - I remember getting it in early early 89' at my local toy store. The Striking Spacey' Box Art was just too amazing
GameBoy
Genesis - Wish i skipped it. It just generates crummy gaming memories outside of a couple of titles.
SNES - More Nintendo magic goodness in 16-bit form. Super Mario World blew me away. Got it a few months after it's launch.
Saturn - Sold it once the PS1 launched, thee' best move i ever made.
PS1 - Favorite Non-Nintendo system Launch period.
N64 - Super Mario 64. Need I say more?
Dreamcast - Appeared more unique and innovative than the PS2. got it on my Birthday a few weeks after it's launch.
GBA
Gamecube - The first Nintendo Console i ever got on Christmas + Luigi's Mansion.
DS Lite
Wii - Got it 1 year later after it's launch on christmas with Super Mario Galaxy. LOVE the Wii.
XBOX 360 - sold it after a year. Thee' Worst Console I've ever owned
PS3- Have the orginal Glossy PS3 in my TV stand as an ornament. looks purty. lol
3DS
Wii U - Still in the box hehe
3DS XL - Shall arrive any day now.

PC- I remember when I was little, playing Curios George Math and Spider-Man Typing. Also I remember loving to watch my dad play Quake II, Lord of the Rings- The Return of the King and Medal of Honor. Not long after I got my SNES, my dad and I both played through Knights of the Old Republic. Still one of the best RPGs ever made.
SNES- When I was about 9 I gound my uncle's SNES at my grandpa's house. This was amazing, because before than, my parents hated video games. But finding it was sweet, because we got DKC 2, Super Star Wars, and Mario All Stars. Still the best console I have.
GBA SP- Got it for Christmas in 4th grade. All my friends were getting the brand new Wii or a DS, but I was super excited to get a Game Boy.
DS Lite- The next year for Christmas, my parents friends all got new DS's and we got their old one, and tons of games for it.
Wii- The next year over Spring break, we got a Wii with Wii Play and Mario Kart Wii on release day. I got so much ridicule from this, because most of my friends think I'm a loser for not having a PS3 to play Black Ops on. Or Skyrim. I'll laigh at them the day good ol' Ninty buys all the gaming companies.
DSi- I got this and for a while regretted it. I gave my GBA SP to my little brother, and thought I wouldn't miss my GBA games. I was sooo wrong. But I discovered Flipnote, and that took my mind off of it.
PS2- All my friends got PS3s and Xbox 360's. My Dad's coworker got a divorce and the husband left his PS2. He was a heavy smoker, so for a while it smelled bad, but we paid 20 bucks and got a PS2.
GBC- I got this to play through classic Pokemon, classically. I got it for $10 and Pokemon Blue for $16.
Darth Vader PSP 2000- This thing is badass. I got this when my friend got a PSVita this Christmas. Needless to say, best free gift ever. Also he gave me a bunch of cool Star Wars and Final Fantasy games.

So there. These are all my systems within my short span of 14 years of life.

I'm going to count both consoles and handhelds (PC is also included) in order of me getting them, and when I was around (look at my username for my birth year ). retro systems acquired from Ebay and alternate revisions of systems are omitted

Super Nintendo Entertainment System - still remember Super Mario World, Super Mario All-Stars, F-Zero, Yoshi's Island, Donkey Kong Country, Tetris Attack, and more
PC - I only played very few games though. I played Monster Truck Madness, Hover, and that retro 6 in 1 Atari arcade compilation. forgot the name of it though
Nintendo 64 - got it with Mario Kart 64, Diddy Kong Racing, San Francisco Rush, and Cruisn' USA (don't judge me). I got Super Mario 64 as well later on
Game Boy Color - like the PC, I only played few games on it. I played Donkey Kong 94 and Pokemon Red
Game Boy Advance - had it around the time it came out. the first games I got for my GBA were Super Mario Advance and F-Zero Maximum Velocity
Nintendo Gamecube - best Christmas ever!!!!!!! I got it with Luigi's Mansion and Super Smash Bros. Melee and I was never disappointed
Playstation 2 - got it for my birthday. Jak and Daxter was the system seller for me XD
Xbox - ok, I will admit this, I really dislike Microsoft when it comes to games now, but at the time I just wanted to try out, but just like the PC and the GBC, I played very few games. it did come with Pac-Man World 2 and some Star Wars game. I also had a demo disc that had Halo 1 on it, and I played that when mom wasn't watching me. I wasn't allowed to play M rated games at the time and I don't like shooters that much now. oh and I liked that one Outrun game, and it was an exclusive game too!
Nintendo DS - the original phat one that came with the Metroid Prime Hunters demo. I didn't play it that much though, until I got the DS Lite and when more games were coming out
Xbox 360 - didn't play too many games on it when I had one either. though I did play Hexic HD and Lost Planet and those weren't bad games I guess. too bad it got the red ring of death, though!!!!!!! not that I care though
Playstation Portable - the only games I really played on it were Mega Man Maverick Hunter X and Mega Man Powered Up. my PSP doesn't work anymore though, but whatever, I prefer Nintendo for handhelds anyway
Nintendo Wii - I got it with Wii Sports, Excite Truck, and Zelda Twilight Princess. I started playing it even more when Super Mario Galaxy, Super Smash Bros. Brawl, and Mario Kart Wii came out though, as well as me connecting my Wii to the internet. such a great system imo!
Playstation 3 - it came with a few games from Gamefly, with the most memorable one being that Ratchet & Clank game. I only watched my brother play it though... I really need to play it more often -_-
Nintendo 3DS - got the blue one at launch, I probably shouldn't have done that though, but hey you can't change the past! anyways, when Super Mario 3D Land and Mario Kart 7 came out, I started playing it even more great handheld so far!
Nintendo Wii U - I yelled out "YES!" right when I unwrapped my Wii U deluxe set for Christmas the first things I've done were set it up, updated it, and played quite a lot of New Super Mario Bros. U. I love communicating with people on miiverse too!

-Commodore 64
-Sega Genesis (first one)
-Nintendo 64 (first one)
-Gameboy Color
-Gameboy Pocket
-Gameboy Advance
-Gamecube
-DS
-Wii
-DSi
-Nintendo 64 (second one)
-Playstation
-Sega Genesis (second one)
-3ds
-Wii U

GameBoy Advance SP
GameCube
DS (original)
I think I got a PC around this time
Wii
Xbox 360 (never use it)
Wii U
(Soon to get 3DS XL and my siblings have some things too)
Not as many as some, I know.
Edit: Actually, I probably got a PC earlier than that.

Nintendo 64
GameBoy Color
PlayStation 1 (the smaller PSone model)
GameBoy Advance
Nintendo GameCube
GBASP
PlayStation 2
Nintendo DS (the original model)
Nintendo Wii
Nintendo 3DS

soon~ Nintendo Wii U

NES = (foolishly) sold it due to unworking anymore
Sega Genesis = (foolishly) sold it already
GameBoy = (foolishly) sold it already
GBA SP = (foolishly) sold it already
PS1 = (foolishly) sold it already
PS2 = keep this one as I realize it has bc of PS1 games and some classic games port range from Atari, Activision, Sega, Capcom, SNK classic games for PS2.
Wii = keep this one as it has bc of GC games, and replaying old classic games from Wii Shop Channel and the WiiWare games too of course.
PS3 = keep this one as I enjoy some PS1 classic games that I haven't played before due to its rare & expensive.

SNES: got it for Christmas the first year it was out
Game Boy: got it for Christmas the next year
Super Game Boy
Another Game Boy. This is the Play It Loud! Green. My original gray console got stolen. I have still have this one, but I haven't played it in a long time. But when I did, I played it so much I wore the serial sticker off.
Game Boy Color
NES. Yep, came from a "one console" house, so I couldn't get one until I was old enough to buy one myself.
N64: Funtastic Green!
Game Boy Advance SP
PlayStation 2
N64. Boring Black. My sister decided she wanted my green one, so I picked up another. Then later she returned the green one. Oh well, at least this spare could be modded to accept Japanese carts.
Nintendo DS: still playing my fat silver
Dreamcast (not sure the exact chronology)
Another SNES (still worked, but lost my original AC and wasn't quite trusting of the third-party AC Adapters to not destroy my original)
Wii
And then the next few it gets bit hazy:
Famicom Disk System
AV Famicom
WonderSwan Color
Genesis with Sega CD
PlayStation Portable
Top-loader NES
GameCube (as I already had a Wii, this was pretty much just for the Game Boy Player)
Game Boy Player
Nintendo 3DS: the only system I bought at launch

NES
Gameboy Color
Nintendo 64
Playstation
Gamecube
Gameboy Advance
Playstation 2
DS
PSP
Wii
PS3
Dsi
Gameboy Micro
3DS
Vita

GBA
GBC
GBA SP
GC
DS
Wii
DS Lite
DS Lite
DSi
DSiXL
3DS
GB
Wii U

NES — My very first system
Gameboy
Super Nintendo
Sega Genesis
N64
PS
Gameboy Color
PC — My very first PC got it to play games like Simcity and Heroes of Might and Magic
PS2
GBA
GBA SP — Traded in the GBA towards the purchase of an SP
Gamecube — Not sure exactly when I got this one it was the year the Zelda Collection disc was included
PS2 Slim model — Sold my first PS2 when I moved so I bought this to replace it.
PC — My second PC need to upgrade from the first one since I wanted to play harder speced games
DS — Mainly bought this to play Dawn of Sorrow at first
PS2 Slim model — Had to get another one cause of the motor burning out in my first one
Wii — Happen to go into Walmart and see one there figured what the hell might as well get it (Glad I did)
PS2 Slim model — Got another one cause the motor burnt out again (Why did Sony make these things so crappy?)
Xbox 360 — Wanted to play some of the game that were exclusive to Xbox and PS3 and Xbox were cheaper than the PS3
PC — My Third PC once again needed to upgrade.
N64 — Had to buy another one cause I sold my first one years ago when I moved
SNES — Bought another one cause my first one was stolen by a roommate
Dreamcast — Bought one at Goodwill saw it cheap and figured what the hell
3DS
Xbox 360 Star Wars Edition — I got it mainly for my son figured he would really like the R2D2 look of it.
Magnavox Odyssey — Bought it at Goodwill to add to my retro collection
3DS XL — Wasnt going to get one at first but than I tried the demo at Target and um yeah.
Wii U

I didnt include addons like the Sega CD since that wasn't a system per say and I didnt include my Smartphone cause I dont use that for games so I dont consider that a system.