Preferably NOT Pokemon, since this is a Pokemon forum and I'm sure a lot of us have grown up with Pokemon, as did I.
I grew up with Mario, Zelda, and in general just a lot of Nintendo games. NES and SNES were sort of before my time, the N64 was my first real console so to speak. Some of my later years when I knew a little bit more about what I was doing, was spent playing Animal Crossing on the gamecube. Good memories. I even remember naming some of our parakeets after townsfolk in the game, lol.