Mario Party is one of my favorites. I have all 4 gamecube version, still need to work on getting the n64 version, and the wii version. Which, the new wii Mario Party 9 is coming out soon, so that's another to add to my list. I loved Sonic, too. Tails is the cutest character ever!
Another of my favorites has to be The Legend of Zelda: Ocarina of Time. I never got to play my own save file, though, until gamecube version came out. [Sibling didn't want me making my own file on her game.] Now I have the 3DS version.