My favorite is Mario Kart 64. This may be partially due to nostalgia (and I probably played it more than any multiplayer game (excluding perhaps mario Party 2 and Super Smash Bros. Melee), but I also feel it was the most innovative when compared to it's predecessor. It obviously kept the same style and concept, but it great improved upon the Super Nintendo one. Double Dash was basically a rehash, IMO, and Mario Kart Wii was basically the same thing, just with updated graphics and a new style controller. I ahven't played the handhelds, so I can't say anything about them, but I feel Mario Kart 64 simply has something special about it and remains the greatest (and most addicting) Mario Kart game to date.
I haven't seriously played a Mario Kart game since Super Mario Kart on the SNES. I've sampled the others. The Wii version is super fun, but the first one will always hold the number one spot for me. Sometimes these 'favorites' have everything to do with when you played them, not necessarily content.
Double Dash!!, hands down. The dual-character setup was brilliant and fun, the special items added a whole new dimension to the gameplay, there were tons of unlockables, and it had the best courses in the series by leaps and bounds. Not to mention the karts.
Wii is number 2, but that's mainly because it had functional WiFi. That's the only reason it beats DS, whose Mission mode was my favorite part of the game.
The Wii version absolutely ruined Battle mode. Forcing teams, no elimination when balloons run out... They killed it. It just isn't fun anymore. The point of balloon battle is to eliminate everyone. What's the point if you regenerate right after? That sucks for the same reason Time matches in Super Smash Bros. suck.