My favorite is the PS3 and Xbox 360
It's between the N64 and PS2. Not sure which.
Edit: Well, if we're counting PC as a "console" then that, but I wouldn't really consider it a console. (Also, a call out to my homeboy the SNES which is the console I started to play a bunch of games on. I played a little bit of NES before, but SNES is where it really started. Also Dreamcast is pretty cool.)
PS2, had so much variety, it showed me games of such different styles... i essentially played like 4 fps's(timesplitters 2 demo, and futureperfect, serious sam. killzone) on that system and everything else was something else. Things like okami, Metal gear solid 3:subsistence, kingdom hearts, god of war 2, dragonball z budokai 3(best dbz fighter ever), and shadow of the colossus. all games that can still be played today and be perfectly playable and still look good.
I will say that this generation is also offering variety to me, maybe because i have the ps3 and pc platforms to play on. I remember earlier on in the cycle i found myself with a very serious case of shooter fatigue that literally almost made me stop being a gamer. If i didnt pick up Bionic commando rearmed during a sale and play that for a month.....i may have missed out. one thing is definite, since then without realizing it, i actively kept an open mind about genres and played games i heard hype about in hopes of "new experiences". Something i didnt worry about in the ps2 days.
I can't really decide between the Vectrex and the RCA Studio 2. The like 3 games they made for both of them combined were three of the BEST games ever. I'm also a fan of the legendary Jaguar and the ill fated Pippin. Talk about Apple never failing. That console was a masterpiece. Though I guess if I really had to choose one the Phillips CD-I has to be it. Those Zelda games were much better than the crap on the SNES. Made the system for me.
In all seriousness, though, I would say the PS2.
After writing this I thought I would look on Wikipedia to see any ridiculous obscure consoles I missed. There was a console called the Ping-O-Tronic released in 1974, before the release of Pong. I tried to read the article on it but it was in Italian. Anyone here hardcore enough to own one of those? If you aren't super familiar with console history that page is quite a read. I'm a game designer and some of the first and second generation ones even I hadn't heard of.
EDIT: Just to clarify I mean the home version of Pong, released in 1975. The Ping-O-Tronic still played a version of ping pong based on the Atari Arcade machine.
If I can choose PC...
But if it has to be a dedicated games platform, then... maybe SNES? Funny thing is that I never owned a SNES, and probably played more Genesis (which I did own), but Mario World and Super Metroid blew me away when I was younger, plus all those *other* games.
Of more modern consoles, I actually thought the Gamecube was fantastic. It was one of the few times that Nintendo actually had an advantage in terms of raw power (Microsoft existed but they weren't really a factor until their system became a Halo machine and then the 360), their 1st party franchises were fresh off the transition to polygons (and some were new to it), they had frickin' Animal Crossing and hadn't run that into the ground yet, a great controller design (and wireless as a option; sucks that you had to forego force feedback), a tiny console with tiny discs and a tiny little handle...
And Wind Waker, probably the last Zelda game I'll ever play and feel like it was a fresh, amazing experience. That's kind of sad.
NES had the greatest effect on me because that was really when gaming as I know it started, so I'd probably have to say it's the greatest of all time to me, but it hasn't held up as well as some of the other consoles from that era. The SNES library is still viable, and it's the only Nintendo system that actually had great 3rd party support, so it would be right there too.