I have one of those.

Home consoles:
NES (Don't remember when, but I did had it)
SNES (Good times...)
N64 (A lot of great games)
Gamecube (Memories...)
Wii (Forever more...)
PS2 (Hate that I owned one of these cursed motherf***ers. I don't know why I owned I twice.)
Xbox 360 (Owned it for 3 weeks, traded it back the next day. Bought when it first came out before I knew about the Wii)

Handhelds:
Gameboy (Like NES, don't remember when)
Gameboy Color (Don't remember when)
Gameboy Advance SP (I still own this ever since it came out. Surprise it still works...)
Nintendo DS Phat (First one got broken. Owned a second one, but gave it to my little cousin)
Nintendo DS Lite (Only for an hour. The battery only lasted for an hour. It was from a pawn shop)
Nintendo DSI (Loved that system, but eventually traded it in for 3DS)
Nintendo 3DS (One of the best consoles, period)

ps1
ps2
psp
ps3
Ps vita (soon)
all purchased in that order idk what exact date.

I forgot that I also used to owned a Genesis and a Dreamcast. Man time flies.

N64 (Recently broken)

Gamecube

PS2

360

Wii

PS3

------------
Gameboy color

Gameboy advanced (brother stole and sold it for drug money)

Gameboy advanced SP (brother stole and sold it for drug money)

Nintendo DS (first gen) (Screen is hanging off the console, but still works)

Nintendo DS LITE

Pretty much had everything at one point or another. Me and my brothers would get different consoles and surprisingly I still have most of them. Skipped DS somehow someone should've slapped me.

Playstation 1
Playstation 2
Playstation 3 -

Dreamcast- Lost
Sega Genesis

Regular Nintendo - No Longer Read Games
Super Nintendo - Gone
Nintendo 64
Gamecube - Lost
Wii

Xbox - Brother Owned
Xbox 360

3DS
PSP
Gameboy,GBColor,GBAdvanced. Lost

SNES
N64
Gamecube
Xbox 360

Gameboy Color
Gameboy Advance SP
DS Lite
DSi

3DS.......when the redesign comes out. As you can see, I always wait for the Nintendo redesign.

Completely personally - not including systems that have belonged to my family/siblings when I still lived with them - I've owned in this order:

1998
+GameBoy Color

2001
+GameBoy Advance

2002
+Gamecube
-GameBoy Color (sold)

2006
+DS Lite (x2)
-GameBoy Advance (sold)
+Wii
+GameBoy Advance SP

2007
+DS Lite

2008
+Nintendo 64
+Virtual Boy
+Xbox 360 Pro
+Atari 2600 (x5)
+Gameboy Advance
+Playstation 2 Slim
+Xbox 360 Core

2009
+Dreamcast
+Playstation Portable
-Playstation Portable (stolen)
+Playstation 2 "Phat"
-Xbox 360 Core (given away)
-DS Lite (x3, stolen)

2010
+DSi XL
+DS Lite

2011
+3DS
+Wii

My family also had two original GameBoy systems and a Nintendo 64.

Atari 2600 - in storage
NES - in storage
N64 - sold it
PS1 - sold it
GC - still have
Wii - still have
X360 - still have

GBA - still have
DS Phat - sold it
DS Lite - still have

Wow, tough question. Really too hard to remember because I'm obsessed with buying redesigns. But I like it, so I'll try.

-Original NES.
-Original SNES
-Top loader NES
-Genesis 1
-Sega CD side slot style
-Genesis 2 (also owned Sega Channel at this time)
-32X (what a mistake)
-3D0 FZ-10 (also a mistake, but I enjoyed SSF2T and road rash and NFS)
-Slim SNES
-Playstation
-Saturn (also own multiple JP versions of this system, love it!)
-N64
-Dreamcast (JP version first, then American, love it!)
-PS2 (never died on me as so many!)
-Game cube (JP first as well)
-Xbox (never actually played it more than once, then sold it)
-PS2 slim
-Xbox 360
-PS3 60 gig
-Wii (Probably the only system I never bought more than once)
-Xbox 360 Elite
-PS3 Slim 120 gig
-Xbox 360 MW2 Elite
-Xbox 360 Slim 120 gig
-Xbox 360 Slim 320 GoW3

I don't know, that's off of the top of my memory, cause I've seriously owned so many versions of so many systems in my day. I owned multiples of the PS1 as well, like the one that had composite cables vs. just Sony's input jack.

I've never owned a Phillips CDi, or Jaguar.

As far as handhelds go, I owned every version of Game Boy, a Game Gear, a PSP, PSP go, 1st DS, DS lite, and a 3DS.

Never owned Turbo Express, Lynx, or DSi or XL.

I still own all of them and have never really had a system fail on me.