Well with the dreamcast you get Shenmue which you mentioned which is an amazing game. You've also got a great collection of Shumps (Ikaruga is on the GC but many of the others are not, like Giga Wing, Mars Matrix, Under Defeat, etc). Then there is the very best version of Street Fighter II (Street Fighter II X), Vampire Chronicles, and Powerstone. I'm also a pretty big fan of Armada and Blue Stinger so yeah the Dreamcast has a lot to offer for me. I like my n64 but honestly there are like 6-7 games on it that are exclusives that I care to play and anything multiplatform I'd prefer elsewhere.

Vita for portable

PS3 for console

Everything else I can possibly get my hands on for their exclusives that interest me

I don't believe in limited choice

Sega MegaDrive CD - purely for Wolfchild OR my master system II - Built in Alex Kidd in Miracle World, Cloud Master, Shinobi. But I'm sure whichever console I didn't go for here I could find the emulators on PC.

I'm torn between 360 and ps3 because I prefer the exclusives on PS3, but there's something about my xbox that I just love. Probably have to go with PS3, unless I can remember the 360 exlusives I love so much (Viva Pinata just isn't enough).

DS Lite (Pokémon, GBA games)

Wii (virtual console, all my GC games)

Well having only owned 4 consoles the most objective answer would be Wii, GC, N64 and MegaDrive.

However a sensible choice would be Wii, PS3, 3DS and Vita... as with various forms of backward compatibility that covers the most games.
I'm not particularly interested in a lot of handheld games though, so I would probably swap Vita for either N64 (to play the great games not added to the Virtual Console), MegaDrive (because Micro Machines isn't on VC) or maybe Dreamcast.

So I think final choice: Wii, PS3, 3DS and N64.

Not the same. I've played both and I still enjoy the originals more. The way the battle systems worked back then, the graphics, the music, I like it all a lot more. I'll admit that nostalgia has a lot to do with it as well.

Playstation 3 (PS3 titles and PSN)
Nintendo Wii (Wii titles and Gamecube titles)
XBox360 (XBox360 titles, XBox titles, XBLA)
iPhone (Covers all previous gens through emulation + plenty of DS titles are available on iOS)

I'll vote this without focusing too much on BC compatibility considering I would just pick newer systems that have a virtual console or backwards compatibility. Instead I'll just vote on which system provided me the most fun during its generation.

1. NES
2. Genesis
3. N64
4. Playstation 1 or Wii

PS2: best console evarr
PS3: 2nd best console evarr
PS1: 3rd best console evurr, rpg console
Vita: Gravity Rush and plays PSP games so 4th best console

1. A JTag modded 360 (loaded up with all NES, SNES, Genesis, N64 and PS1 roms)
2. GBA for the best handheld gaming experience ever
...
And that's all I need

I've owned an NDS. It's good, but it lacks variety, much like the SNES. I can only have room for one lacking console. Vita can play a lot of PSP games, as they're available for download and can also play a lot of things the psp can't.

If you owned a Vita, you would understand. It really is potentially the (pronounced thee) best designed console, next to the SNES. Everything just feels perfect. It'll take some time for it to get a bigger library, but it really is the cat's pajamas. I  feel the same way about WiiU. I think it's potentially one of the most unique gaming experiences to ever hit, and the idea is really solid. I like the Wii a lot, but it's just lacking "something"...it's style ovver substance. I believe that something is the wiiU pad.