The first games we ever owned were Pajama Sam, Sonic 3, a few Need for Speed games, and (later) the first two Harry Potter games, all for the computer. I got my GBA when I was eight or so and had Pokemon Crystal, Super Mario Bros. 3, some racing game whose name I can't remember, Shrek 2, SpongeBob: Battle for Bikini Bottom, Mario Pinball Land, and a couple more. My first "real" system was the GameCube, and after a couple of years of owning it I wasn't really a kid anymore.
I also remember playing Pokemon Blue at my cousins' house and, even earlier than that, seeing PS1s in stores and asking my dad to get us one but him refusing. I do own a PS1 now but didn't until a few years ago.