Hard to say my favourite gamecube game, so I shall give a list of some of my favourites and say why.
Also, the Wii is my favourite console because it can play the AMAZING games on Gamecube, and also download older games (or newer ones, but I like Virtual Console titles mostly). Also some Wii games are pretty awesome as well.
In no order,
1. Kirby's Air Ride - Amazing and unique racer with the adorable Kirby! The best part is City Trial mode, you collect power-ups, fight your friends if you want, find new vehicles, explore the city, and sometimes fight bosses, then do an event which can be a giant brawl, a race, target practice or how far/high you can go. It's sort of like Smash Run. And yes, you can (sort of) fly.
2. Smash Melee - Do I need to explain?
3. Mario Kart Double Dash - Do I need to explain?
4. Sonic Adventure 2 Battle - This was the first version I played, and it was awesome. Awesome music, awesome gameplay, fun multiplayer, and cute Chao! :D
5. Paper Mario : The Thousand Year Door - Possibly my favourite Mario game (definitely my favourite spin-off), it was one of the few RPGs that had gameplay that I actually found fun, and along with a good story, good music, and great character writing, and design, you have a great game.
6. Sonic Heroes - Not an exclusive, but a great Sonic game, great music, and great multiplayer.
7. Animal Crossing - New Leaf and City Folk were definitely better, but this was a great start to the series. It's a simple and calm game.
8. Pac-Man : World Rally - Another non-exclusive, but this title isn't as big, but is still great. It's pretty much Mario Kart, and I like this game better than Double Dash. Just my opinion :)
9: Super Monkey Ball Adventure - People don't seem to like this one, but I love it. It has the same great gameplay, along with an awesome multi-player mode with cool monkeys, and a neat over-world! The only problem I have are the loading screens take a while, but not too long.
10. The Legend of Zelda - Windwaker - It's a great adventure with great graphics, and the only Zelda I like
11. Wario Ware, Inc. Mega Party Games - Most people don't know or talk about this one, but it's a fantastic party game, or single player game. It's like the later games, but simpler.
I wonder if anyone actually read that :P