I know it's been said before, but the very first thing that popped into my head was Bioshock. To me, the fun of Mass Effect and Fallout is tweaking character builds to find the most awesome one for you. If you erase what you know about your builds, you're just going to awkwardly play through it again. Sure, the story is nice, but they're loose - choose you're own adventure - types of stories and honestly, if you like that so much, go play D&D or something.
But Bioshock, that was a joy. I saw the plot-twist coming from six miles away, and they still managed to surprise me with the details of the very twist I saw coming. Really anything with a tight and fantastic story. So, on this list would be:
Bioshock
Psychonauts
Final Fantasy Tactics
Halo (Yes, I said it, I loved the original story)
And if I could: The Star Wars Original Trilogy so I could watch that again with new wonder.
But... it's only one... and even though Tactics is still my favorite game ever, it'd have to go to Bioshock because that story is just amazing.