In my earliest days, I played games off of a DOS-based database computer (I don't really know what it was, I just know that you directed its actions with a command prompt XD) and I beat Chip n Dale Rescue Rangers. Very proud moment in my young life. There was also this other game, probably an 8-bit, called Castle, where you were a small spade-shaped icon running around a "castle" collecting items to aid your escape. I never beat that one, but got very close, and think I know how to beat it now XD Unfortunately I can't find the floppy that had the game on it, nor can I find it for download anywhere. (But I did find Zork... oh, how much fun is Zork. Kinda annoying, but you can play it on Uncyclopedia too... they have some very funny versions of the game there. But I digress).
Anyway, what I would really consider the first game I beat was The Legend of Zelda: Link's Awakening DX. It was super-difficult for me, too, because I had no real concept of how an adventure game like that worked. I'll admit I had to consult a walkthrough a couple of times, too. Still an excellent game though, and a favorite of mine until this day ^_^