Hmm I'm not sure if it started with the GameBoy or GameCube.
My friend had Pokemon Yellow on the original GB. One day he decided Pokemon was un-cool so he gave me his GB with Pokemon Yellow!! So pretty much since then I was heavily into Pokemon but not much else.
I decided to buy a GameCube when the Sega Dreamcast failed as I knew Sonic was heading that way and of course Pokemon would likely end up on GC too. But the GameCube come with 2 titles I'd never played before. Zelda: Wind Waker (incl OoT) and Super Smash Bros Melee! I remember playing Wind Waker for the first time and being completely blown away. That was probably when I become a real Nintendo fan.