It has been suggested, and then seconded, and suggested again and again. Pound for pound Final Fantasy 3 is IMHO, the greatest video game I have ever played. Years after no one was playing with a SNES it was re-released for the Playstation, and was extremely re-playable even after all those years.
Final Fantasy III (now known by its Japanese name as Final Fantasy VI) was also released for the Gameboy Advance a couple years ago, and is still available in stores (and can be played on a DS).
In addition to what everyone else has mentioned, the Super Star Wars trilogy for the SNES (Super Star Wars, Super Empire Strikes Back, Super Return of the Jedi) is fucking awesome. The games are hard as hell, but very enjoyable and still look great.
The SNES had some great fighting games, too. The system got the best 16-bit versions of Street Fighter 2 Turbo, Mortal Kombat 2, and Teenage Mutant Ninja Turtles: Tournament Fighters. There’s even a port of Street Fighter Alpha 2, although I’d imagine that’s hard to find and pricey now.
I loved NBA Jam: Tournament Edition, and I didn’t even like basketball then.
(Remember this: Down R R B A Left)
I second FF3 and SimCity.
This, This, This, this, THIS!
Oh oh, and don’t forget Harvest Moon, the original one for the SNES. It’s a classic. If you’ve never heard of the series, you play a farmer who’s just been given his grandfather’s old and rather shabby farm, and it’s up to you to clear the field of weeds, rocks, and old tree stumps, then grow your crops to generate cash for things like cows and chickens, which will add to your income as you grow more crops, and use your money for supplies, extending your home, etc. Different seasons have different crops to grow, too.
On the side, you can take part in festivals and even court several girls to eventually become your wife, then have a baby.
The game ends after about 2 1/2 “years”, and your parents tell you how well you did. It’s actually pretty fun to play.
A lot of good games suggested here–brings back memories. 
One that I didn’t see mentioned yet (at least while skimming through the other responses) is Populous. I remember spending many hours with this game when I first got it. Deceptively simple, but very addictive.
The original Civilization is available on the SNES. IMHO, the interface was vastly preferable to the PC version of that game. It is actually the game that got me hooked on the entire genre of 4x games. I recommend it, if only to see where the series started.
Did they have Zombies ate my neighbors for SNES? I know they did for Sega and that is the single best game ever made. EVER!
Indeed they did.
Firestrikers - I enjoyed this one as well, though I was never able to beat the last boss.
Also Lemmings is a good game for the snes, but I’m not sure how treadmillable it would be
Yes (although it was renamed Zombies for the European market). It was better, too - the SNES pad worked a lot better for that game than the 3-button Sega one.
The imagined sound of chainsaws kept me up many a night in those days. Game also notable for first 16-bit upskirt…