I share a fair number of the ones already mentioned (FF 3/6 especially, but there are a few old favorites for the SNES I get to mention first:
Earthbound: I love this game. With quirky mechanics (roll-down hp meter, the ability to order a time-delayed Pizza of Healing before setting out to the dungeon) and an even quirkier plot ("Please, will you sign my banana? I need it so the museum curator will let me into the sewers to fight a Rat of Timeless Evil." And who can forget the Pencil Eraser? And the improved version, the Eraser Eraser?) this game was one of the most memorable RPGs ever. I lent it to a friend and for a while afterwards we got into the habit of confusing our other friends by reminiscing about Major Psychic Psychos and Happy-Happyists. Any game that forces you into a deadly duel with the Unassuming Local Guy is classic.
Secret of Mana: I almost bought the old SNES 4-person converter thingy for this. Boss battles tended to be too easy unless you were careful about limiting your magic use, but the plot was solid, the weapons were fun (especially the boomerang tree and the glove tree), and this is the only game I didn't really mind leveling up in. Great stuff, especially for being in the ChronoTrigger genre. I also liked Secret of Evermore. Loved the dog, loved the alchemy, loved the characters. Drops a rank behind Secret of Mana because of some annoying labyrinths (a hedge maze, looping palace, and confusing space station all in one game? Aargh!) and because I was never able to beat it due to an annoying glitch.