I'm sure I'm missing some greats, seeing as the GameCube was my first actual Nintendo Console. However, I've had a GBC, GBA, GBA SP, and a DS, so handheld wise I'm good to go.
1. Paper Mario: The Thousand Year Door (GC) - Definitely my favorite, Mario RPGs are always creative, funny, and do stuff for you that other RPGs don't. The worlds and design's are very creative and unique.
2. Pokemon Gold/Silver (GBC)
3. Mario Kart DS (DS)
4. The Legend of Zelda: The Wind Waker (GC)
5. Super Smash Bros. Melee (GC)
Almost in the Top 5:
Golden Sun (GBA)
Animal Crossing (GC)
Paper Mario (N64)