Everyone is going to say Final Fantasy and Chrono Trigger, so I'm going to talk about some of my favorite SNES RPGs that don't mentioned often.
Soul Blazer - It may not be the more flowing and faster paced game that Illusion of Gaia, its successor, was, but there's something incredibly pleasing about the methodical, exacting gameplay in Soul Blazer. The town building is also incredibly rewarding. Then there's the music. Wonderful.
Dungeon Master - SNES saw a number of high quality and, frankly, superior ports of great PC RPGs. Dungeon Master is one of them. It remains one of the most underrated and highly influential PC RPGs. It took Wizardry's mold and injected it into real time, an evolution that would go on to influence Ultima Underworld. It's also the first in a line that would lead to Eye of the Beholder and then on to Legend of Grimrock.
Ultima VI - Another wonderful PC port. It outdoes the PC game in music, in having a full screen display (PC had huge UI boxes, with the game in only 1/4 of the screen), and in playability (controller > primitive mouse controls). All that while cutting very little content (though the dialog system does suffer). Beyond that it's still one of the best games in one of the most interesting RPG series out there. It did open world long before anyone called it open world. The world was richly designed with unique quests, treasures, and monster encounters everywhere.
Shadowrun - One of my favorites. It's one of the best action RPGs on the SNES, with fun combat, FLOATING COMBAT NUMBERS, a great story, great pixel art, and great music. Does it hold as closely to the Shadowrun ruleset as the Genesis game? No, but I enjoy it far more.
