The first game I ever beat was Super Mario World, the original for the SNES. I started playing the game at around age 6, when I turned 7 I managed to defeat the game. Around that period of time I also beat Donkey Kong Country for the SNES.
Final Fantasy IV for the Super NES. I beat the game in just under twenty-two hours, even clearing all the secret areas and doing most of the side-quests that some of my friends told me about. Also, I enjoyed it so much that I bought it the next time I saw it out on display at the grocery store that I initially rented it from. It's my oldest game next to Super Mario World and The Legend of Zelda: A Link to the Past (both of which I didn't beat until much later on...).
I dunno, but the first game I remember beating was Yoshi's Island. I played that game religiously (got perfect scores on every level and such). I still play it today sometimes.
And when that old-game-collection comes out for the 3DS, I'll probably get it for Yoshi's Island :>