I used to watch my grandma play Yoshi's Island on SNES all the time, apparently. I don't really have any memories of it myself but she told me about the time she finally managed to get to the end (I don't think it had a save function?) and I, about two years old, got scared of Bowser and turned the game off. (In my defense, he looked like THIS!)
My mom and I also played on the Sega Genesis all the time. Mostly Sonic 2 and Bubsy.
I also used to play computer games a lot. I remember Disney Magic Kitchen and a My Little Pony game.
I got into AC around the time it first came out though, when I was about 7, so I count it as one of my favorite childhood games too. My other favorite series was and is Harvest Moon, which I got into with More Friends of Mineral Town when I was about 9. For the GBA I really liked that, and Kingdom Hearts: Chain of Memories (even though I knew nothing about Kingdom Hearts) for some reason.
EDIT: Omg, actually how could I forget about Tomb Raider? My dad and I used to play that all the time when I was like 4-10. Mostly just me watching him play or pressing the spacebar for him. For years I wore my hair in a braid because I loved Lara Croft so much... When I got a little older (~8) I played by myself, except until I was in high school I only ever played in the training level (her house) because I couldn't stand seeing her get hurt.