Shigeru Miyamoto and his team at Nintendo of Japan sure hit it out of the park when they moved Mario into the third dimension, didn't they?
I remember being absolutely amazed by Super Mario 64 when it was unveiled at Nintendo Space World in 1995. A year later, after receiving a Nintendo 64 and a copy of the game as a birthday gift, I was even more amazed as my college roommate and I played through it during a wintry Thanksgiving weekend.
If that doesn't speak to the game's accessibility, I don't know what does.
I didn't include Super Mario 64 on this list because it's accessible, though; I included it on this list because it's enjoyable. In fact, I'd go so far as to say it is one of the most enjoyable games--if not the most enjoyable game--I've ever played.
The aforementioned roommate and I played an awful lot of Super Mario 64 that year, and the next year my new roommates (all five of them) and I played it even more--and all the while we had huge smiles plastered on our faces.
Sure, we liked how the game controlled and looked and sounded (especially that wacky, hillbilly-meets-Mario "Secret Slide Theme"), but what kept us coming back for more (and more and more and more) was that the game was fun with a capital "f."
See also: all of the previous '10 video games that made my life gayer' posts