The SNES was my introduction to gaming. I'm not sure if you want the obvious classics pointing out. I must admit I played more action games as a kid rather than turn based games or RPGs but I've played more of the later since.
EVO Search for Eden is an interesting RPG-like action game where you have to eat to gain XP which is used to evolve your creature. It has many evolutionary branches to explore. I don't want to say too much.
Super Mario RPG is an interesting take on the RPG genre and is much like the more recent mario RPGs.
Chrono Trigger is a must if you've never played it. RPG again. Might be worth giving the DS version a try. (ninja'd)
Terranigma is an action RPG never released in the states.
Secret of Mana is a fantastic game, best with two players, though.
Tactics Ogre: Let us Cling Together is a turn based, isometric, tactical RPG not unlike the more recent final fantasy tactics games.
Mother 2/ Earthbound is regarded as a classic in the turn based RPG genre.
The Legend of Zelda: A Link to the Past. One of, if not the, best Zelda game. Plenty of dungeons.
Also do you know any good Star Wars console action/RPGs?
The super star wars games aren't bad. One for each of the original films.
Other games I'd suggest which don't fit your description:
Donkey Kong Country 2 was and is one of my favourite platformers. Fairly lengthy game with great music. The other two are fine too.
Flashback is like Another World and Prince of Persia if you like that kind of thing.
Starfox 1 and 2. Your mileage may vary as by today's standards they are pretty basic. First - Third person linear / open space ship fighter games. You'll need to get a patch for the sequel as it was never released, let alone in English.
Super Mario World and Yoshi's Island are both classic platformers.
Ghouls 'N Ghosts is available on almost every console. An interesting take on platformer mechanics.
Super Metroid
Sunset Riders is a blast and especially with a friend. Side scrolling cowboy shoot-em-up.
Smash TV is an arena shoot-em-up. It's like a violent future game show. You can win a nice VCR!
Metal Warriors is a short well made mech platformer with multiple mechs and the ability to leave yours. Nice animated cut-scenes.
There are still more I could mention both RPG and the rest...