BIG SNES fan here!
I've played all the obvious and popular ones like Mario and Zelda. Anybody played Kirby? I loved playing that little pink fella
I'm surprised nobody mentioned Chrono Trigger. It was made by Square Enix and it's an RPG. As far as I can remember, you have six characters and each comes from a different historical era. It's a very nice game, especially considering RPG wasn't as popular back then.
My guilty pleasure was a game called Turtles in Time. It's the notorious Teenage Mutant Ninja Turtles but they game is a beat them up type. I was in love with Raphael (the red bandana) as a kid so I played through this game way too many times.