(Megavolt, I'm lookin' at you!).
- Secret of Mana (I never played it. Yes, seriously)
That's the game that single-handedly launched my RPG addiction, so you're in for a treat. I was basically floored with that title sequence and music as a kid. "Welcome to the world of Square RPGs", and all that.
And I second Csk's great list, except for Final Fight 3, which is a game that I feel had a lot of potential but was let down in particular by the horribly frequent reuse of the same set of enemies over and over again. I played the game with my sister and we were almost laughing at how most of the time you're just fighting variations on Andore. Beat 'em ups are by nature somewhat repetitive, but I felt like FF3 took it to a new level, and it's a shame because the controls and movesets for your characters are great, which is one thing I do agree with Csk on. Yet Captain Commando, an older game, has a much better variety of baddies along the way. The first Final Fight for the arcade or the Sega CD is the way to go.
Wild Guns is the kind of game that is good to throw at the Genesis fans who say that the SNES was weak in terms of action games. Say what?! The experience is totally unique, stylish, and fast paced. It even supports co-op! It seems like Natsume could do no wrong with the games they published for SNES.
Super Punch Out gets overlooked because it didn't have the same impact as the original. Instead it's something of an improvement on the original formula. Either way it's a wonderful game that is easy to pick up and play at any time, and trying to beat your own best times for the various fights gives it depth and tons of replay value. I personally like it more than Punch Out.
Oh, and Ninja Warriors is definitely good. I was a little apprehensive about the left to right style of the game but the stages are nicely varied and each ninja possesses a cool set of moves.
I'll just edit/post a list that I had already been working on for a thread on another forum: (excluding imports, of course)
Final Fantasy IV & VI, Earthbound, Chrono Trigger, Super Metroid, Ogre Battle (some people are turned off by the strategy element, but I love this game; it was basically my first SRPG), Zelda 3, Super Punch-Out, Super Mario All-Stars, Super Mario World 1 & 2, Super Mario Kart, Super Castlevania IV, Actraiser, Soul Blazer, Illusion of Gaia, TMNTIV: Turtles in Time, TMNT Tournament Fighters, Pocky & Rocky 1/2, Demon's Crest (stick with it and remember that using curative items and finding new powers does much to alleviate its difficulty), DKC series, Space Megaforce, Axelay, R-Type III, Gradius III (slowdown or no, it's a fun game), Zombies Ate My Neighbors (this game is too awesome), Contra III, Arcana, Breath of Fire 1 & 2, Lufia 1 & 2, Secret of Mana, Secret of Evermore (yes, it's a good game with some nice music from Jeremy Soule), Super Mario RPG, Rock 'n Roll Racing, Plok! (the Follin brothers handle the music in this one), Street Fighter 2 Turbo, Killer Instinct, Mortal Kombat II, NBA Jam T.E., Megaman X1-3 (two and three still fetch a pretty penny, but the original is dirt cheap), Megaman 7, King of Dragons, Knights of the Round, Saturday Night Slam Masters, WWF Wrestlemania the Arcade Game, Wild Guns, Battletoads & Double Dragon, Super Turrican 1 & 2 (part two deviates from the Turrican style and opts for a more Contra-esque style, but unless you're one of those diehards who feels betrayed or whathaveyou, the variety is impressive), Tetris Attack, Sim Ant (I like the SNES version more than the PC version), Super Bomberman 1 & 2, Dragonview (don't judge by Drakkhen; the sequel is an excellent game), Soldiers of Fortune, Cybernator, Metal Warriors, Starfox, Super Star Wars trilogy, EWJ 1 & 2, Cool Spot, and others that I'm forgetting.
Sorry if it's messy, but organizing it line by line would've made it too long. Oh, and I've always liked Taz-Mania and Jurassic Park. I just wish that Jurassic Park had a password feature. Prehistorik Man is good too (although it has Banjo Kazooie-type "voices", heh heh), though I'm not sure how it relates to Prehistorik 1 & 2 (whether it's a sequel or a port of one of those), which I've never played. First Samurai for example appears to be a port of the Amiga game.
Terranigma is obviously a great game (the last part of Quintet's Soul Blazer trilogy), but there are definitely some potential issues with trying to play it on an American SNES. I think that some copies of the game have a region lockout chip that will block your access even with a converter or without the tabs in your SNES. Or at least mine does.