My favorite GameCube game is Fire Emblem: Path of Radiance. It was my first Fire Emblem game and I find the story, characters, and objectives to be some of my favorite in the franchise. It also led to a direct sequel for the Wii, Fire Emblem: Radiant Dawn, which I also adore.
That said, there were a lot of good games for the GameCube and I find it to be a pretty solid console. Animal Crossing, Kirby Air Ride, Luigi's Mansion, Mario Party 4-6, Super Mario Sunshine, and Super Smash Bros. Melee were all games I grew up with and enjoyed. Though I'm not really into the Metroid or Zelda franchises (I'm content just watching others play) there were also Metroid Prime 1 and 2, The Legend of Zelda: Wind Waker, and the original version of The Legend of Zelda: Twilight Princess which seem to be highly regarded by fans.
TBH, Pokemon Colosseum and Pokemon XD. I was surprised at how fun those games were. I no longer have a Gamecube or those games, but I did put a lot of my Pokemon from those games into my DS games (put them in GBA, then transferred them up to DS).
I would say Wind Waker but the HD remake outclasses it in so many ways, so I'll go with Air Ride. One of the few GC games I still regularly play. Chibi Robo would be second.
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Oh yeah the Metroid games were also amazing but since they were in Trilogy, I wouldn't really count them either. (I just now realized how many GC games were remade...)