Devil May Cry 3 was the game that really got me hooked on playing games seriously. Its gameplay is just so responsive and has an incredible amount of depths. I still don't think its gameplay has been topped.
I grew up watching my brother play the SNES though (Castlevania, Zelda, Street Fighter) and played some games from that era, but I didn't start to play games as one of my main hobbies until much later.