Four. I completed Super Mario Bros. on our NES. It was one of the three games we had (Excitebike and Duck Hunt being the other two).
My sixteen year old brother was pissed, because he still had not been able to finish the game (and a four year old beat it first :lol

.
EDIT: I think my nephew has me beat though. He finished Pokemon Blue (with all eight badges/Elite Four win and a ton of Pokemon) before he could read. And for a game that is a ton of reading, I thought it was pretty damn good.
EDIT 2:
[quote name='javeryh']Anyone who thinks they beat a game at 3 or 4 is just crazy. I am around 3 and 4 year olds all the time (I have a kid that age and so does everyone I know) and they just don't have the reflexes and dexterity necessary to do something like that. I mean
maybe a 4 year old could finish a game but he or she would have to be just shy of 5.[/quote]
I think it may depend on the child, and what their surroundings are. My siblings were much older than I was, and so, I had nobody to play with while they were in school. Naturally, games were a good alternative until they got home. And that's mainly why I remember when I finished SMB. It was before I was in school, and my mom actually wrote it down for me (and the note was tucked inside the SMB case at home

).
That being said, another young nephew, is making his way through SMB on the VC. He is four right now, and will be turning five in August. He isn't through it yet, but for his age, he is extremely good. (I think he makes it into World 4 or 5...but I haven't showed him where the warp zones are.)
Another thing I believe that helps is "pure concentration." I used to get red in the face while playing, and it was because I focused on nothing but the game. As you get older, you multi-task, or listen to other things while playing (music/people/etc.). My nephew seems to be able to ignore everything else but the game (like I did...haha).