Excluding Wii U since it still has lots to offer, what is your favorite Nintendo console and why?

Even though I have had all of them I will have to say the Gamecube.
I guess because at that time I was exposed to a lot more genres and experiences. Plus I was able to afford more than a handful of games a year.

Still I have fond memories of playing multiplayer games with family and friends, and also enjoying a slew of titles that I became really immersed in.

Plus gaming on Gamecube I really played games for what they were and not what media told me to (the dark days of modern gaming....).

All hail the virtual boy.

SNES. My favorite console of all! Awesome games like Super Mario Kart, Super Metroid, Super Ghouls 'n' Ghosts, Final Fantasy II and III, Secret of Mana, Super Mario World, Yoshi's Island and many more are still some of my favorite games today.

N64 - OOT, Mario 64 and Goldeneye are among my favorite of all time.

3DS

LOL! This! So much this!

But for the OP, it's a tough call for me between N64 and Wii.

Interestingly enough, I think I give the edge to Wii. It was the first system I bought with my own money (waited 13 hours in line to get it at midnight on launch day--for Twilight Princess, of course!). It has my favorite game of all time, Xenoblade Chronicles, two fantastic Zelda games, two great 3D Mario games, two great Kirby games, a great Donkey Kong Country game, Smash Bros. was still great, the Goldeneye remake, Rayman Origins, The Last Story, and then I had hours and hours of fun playing "casual" titles with people/friends who had NEVER gamed before. The likes of Mario Kart Wii, Wii Party, Wii Sports, Wii Sports Resort, and Wii Fit got so many of my friends and family members gaming for the first time. And they wound up liking gaming so much that they would then take steps forward and play games like Super Smash Bros. with me as well. 

Snes

On that list, I have only owned the GameCube and the Wii. I gave it to the GameCube, only because it had Pokemon Colosseum and Pokemon XD: Gale of Darkness. The Wii, however, was a close second.

Gamecube. Had my favorite Zelda (Wind Waker), and my favorite Mario (The Thousand-Year Door), plus other great games like Super Mario Strikers, Pokémon XD: Gale of Darkness, and Luigi's Mansion.

I still consider TTYD to be my favorite Nintendo game. It's a shame they had to go and change up the formula for the next two entries in the franchise, especially Sticker Star.