First game I remember:
Playing a friend's copy of Pokémon Red. Many a late night was spent getting as far as I could without being allowed to save.
First games I remember owning:
Sega Genesis 6-PAK with Sonic the Hedgehog, Golden Axe, Streets of Rage, Revenge of Shinobi, Columns, and Super Hang-on. Mostly played Sonic. Mom forbade us from playing the "violent" games, so we'd get in trouble if we were caught playing GA, SoR, or RoS.
First game I have fond memories of:
Spyro 2: Ripto's Rage. This was one of the first games we got when we upgraded from the Genesis to a PS2, and I played it many times over from the start before we invested in a memory card. The first game I actually beat start to finish without saving. I fondly remember 100%ing it over a weekend, leaving the PS2 on overnight while I slept.