Dark Law: Meaning of Death is a good SNES RPG, though it is a Japanese only release (with an english translation patch). Combat is turn based and is apparently similar to Treasure Hunter G (which I've never played). For those of us who have no idea what that means: each character has a set amount of "action points" portrayed in a bar above their sprite that drains as you move them around and you can only take other actions (attack, cast a spell or use an item) if you have enough AP left. The higher agility a character has the more AP they have.
When you first load up the game you can create up to four characters, two of which can be male, and the other two female. The males start a little bit stronger, but the females level up quicker. The character's base stats are the same every time you select them, however the bonus points that you can spend can range from 4 to 12 (that I've seen). If you're like me and you like to abuse "random" seeds in games to get the best results all you have to do is select the character, give them a quick name (just "A" works fine), see how many bonus points you get and create a save stat if you're unsure whether you'll keep that bonus or not then back out to the main selection screen again and keep trying to get a large amount of bonus points. Once you're satisfied you can back out to the name selection screen and name your characters whatever you want (the seed only changes when you back out all the way to the main screen so it's safe to alter the characters name).
A similar trick can be used when leveling up, which can be advisable since in this game
leveling up can actually lower one or more stats. In this situation you can randomize the seed by pressing the "Luck Shuffle" button in the System tab of the game menu in between attempts to level your characters up. The same button can be used to change which item you get out of a chest (each chest carries two different items). Using a spreadsheet can help a great deal when leveling up if you want to obsess over getting the best stats possible lol.
Dark Law also features something of a class system in that you can visit the Guild and change your characters Jobs between things like Merchants, Town Guards, Fishermen, Bards, Thieves, Clowns and so on which influence how your stats change at level up (I believe) and provide various skills upon leveling up (randomizing the seed can help you learn skills as well as they aren't set for you to learn them at specific levels).
It's a challenging game with good music, graphics and an interesting quest system. As you visit the various locations in town you can talk to people and receive quests that allow you to travel to various scenarios. There is also a large dungeon that you have access to at all times which allows you to earn gold, experience and, if you don't get lost, find treasure chests that contain powerful equipment. Certain skills like lock picking are essential for exploring that dungeon and getting all of the chests.
And of course there are the Dragon Quest games, all of which are Japanese only (if you want to play the SNES versions). Of the three Dragon Quest III is my favorite by far, mostly because of the customization available to you. At the start of the game the hero (or heroine) is speaking to Yggdrasil (the tree of life), or God, or some powerful being who asks you a series of questions which determines your characters personality in game. The main character's personality has a heavy influence over her stat gains at level up, and females come up on top for two reasons. They get the most powerful and balanced personality (Sexy) and they get some of the best equipment, though you aren't going to be screwed over if you choose a male character instead, you just (probably) won't end up with as high of stats as a Sexy female.
DQIII is often accepted as the most difficult Dragon Quest game since it feels like the game expects you to take use of Dharma Temple and switching classes to fill out your spell lists and strengths while covering your weaknesses.
The SNES version of the game is a remake of the NES version (which was released in the US), and was followed by a GBC port/remake (which was also released in the US) though the SNES version is by far the superior version despite the need to hunt up a translation patch.
Dragon Quests V and VI are of course worth playing as well if you enjoy the series.
And of course I can't recommend SNES RPGs without mentioning Earthbound (and shame on the rest of you for not doing so already) which has similar gameplay to Dragon Quest but one of the best stories and soundtracks of any JRPG. I will warn you though, even if you love 2-D gaming the graphics may take some getting used to, despite being a SNES game it almost looks like it's for the NES
but once you see the humor in the game the graphics design fits the game perfectly, especially the enemies.
I'd elaborate but I'm tired, so check out
STARMEN.NET for pretty much everything you'd ever want to know about the game, and more (including a walkthrough).
I'll post more later if I remember. Also I fully endorse the recommendations for Terranigma and nacho is right, the game had a Japanese and European release, though oddly no American release so you can play it in English without hunting down an English patch.