There are a couple of games I can remember that I played when I was around 4. I never beat them, but they are some of my earlier memories:
The Fantastic Adventures of Dizzy (I found it terrifying yet awesome at the time; my mom used to play it with me)
Super Mario Brothers
King's Quest (I don't remember which one, but I got stuck when I was supposed to present an item to a dwarf with large eyes)
The first games I really delved into and gained a real appreciation for the medium were later on when I was a tween:
Suikoden
Final Fantasy Tactics
Tony Hawk
Silent Hill 2 (This was one of the first horror game I can remember playing; I quickly got overwhelmed and terrified when I first tried it at 11ish and gave up. Thankfully, I rented it again years later.)
Star Craft (I was so awful, but I loved it and wasted tons of hours on it. It was the first game I played on the internet.)
It occurs to me that there are tons of other titles that come to mind like Conker's Bad Fur Day, Dance Dance Revolution, Army Men, Mario Kart, Final Fantasy VII, Yoshi's Story, Doom, and Fatal Frame. I have a lot of good memories attached to some of these; I find that while a lot of the fondness comes from the content of the game itself, some of these games have specific memories attached to them like who I played with or who else was engaged in watching (and occasionally back-seat gaming) at the time.
As a kid, I was annoyingly excited to introduce others to new games (especially horror), try ones my friends/brothers liked, and even watch others play. I think that, at least to some extent, really anchored me as a gamer. Does anyone else notice a social element with their earlier game experiences?