I would say it's superior to FF4 and FF6, but not FF5.

Ah, Bahamut Lagoon. Good game, but Yoyo is a terrible character.

If we're recommending strategy RPGs, then Fire Emblem 4 (Seisen no Keifu) and 5 (Thracia 776) are way up there, too. Last I checked, only 5 has a full translation, but both are worth playing. They're a serious gateway drug to the rest of the series and have some seriously amazing characters. There's also murder, suicide, incest, possession, child sacrifice, and all kinds of other dark stuff. Game of Thrones presented by Nintendo, basically.

Yeah, I remember something like that happening not so long ago. Isn't Mother 1 a Famicom game of the non-super variety, though? Not sure why it's being brought up.

Mostly already mentioned but I'll add my recommendations:

Seiken Densetsu 3 - One of the best RPGs ever, such a damn shame it was never properly localised and released in the West...

Final Fantasy 5 - My favourite FF of all time, fantastic job system and pretty decent story.

FF 6 - Not quite as good as 5 (I think) but a damn good game anyway.

Chrono Trigger - Absolutely great RPG, fun characters and setting. A classic.

Bahamut Lagoon - A great tactical RPG with interesting Dragon evolving mechanics (not sure it was ever properly localised over here)

Illusion of Gaia - Quite different and fun game.

Super Mario RPG - Funny Mario based RPG.

Not technically RPGs, but great games:
Actraiser 1 (not 2, as already mentioned the second ruins the formula), Link to the Past

Final fantasy 6
Breath of fire
Super mario RPG
Lufia 2
Secret of mana
Ogre battle(Strategy RPG)
Front mission(Strategy RPG)
Tales of phantasia
Star ocean
Illusion of gaia
Terranigma
Brandish
If you can find the translations,try out these as well:
Seiken densetsu 3
Romancing saga 1,2,3
Langrisser
Bahamut lagoon
Dark law:meaning of death

Can't say I remember too much about it (I have to check a wiki any time I mention it as I mix it up with Terranigma) but I remember it being RPGish....

Though thinking about it it may be that it's more Zelda-like than RPG...

sega genesis shadowrun was better.

There is only one.

Behold Final Fantasy V.

Okay, it may not be the only one, but its the most superior SNES/SF RPG you'll ever find.

...But if that doesn't appeal, try Ryusui's fan translation hack.

Star Ocean and Tales of Phantasia

I'm going to throw my hat in the ring for Sailor Moon: Another Story. Yeah, it's basically just replaying the story of the anime, but there's an original narrative going on around it that justifies you replaying the story of the anime.

SNES was definitely the pinnacle of RPGs. A pity the genre has fallen so far... But then again, I suppose most genres have declined overall, but none have fallen as far as RPGs have.

Live-A-Live - which is probably the best game I've ever played
Final Fantasy IV-VI
Super Mario RPG
Bahamut Lagoon
Secret of Mana
Mother 2

tales of phantasia, terranigma, chrono trigger and final fantasy 6, all have awesome story that will stay with you forever

For Super Nintendo, Super Mario RPG had been an EXCEPTIONAL title, deemed by many players as strangely underrated beyond belief, silent hit, gem, etc. For some reason, it is very well made and playing it make you stick with it in awe. Besides breath of fire series, it is the one and only turn based RPG that i could "stomach" and i actually bothered to finish (plus find all secrets as well) and this is something, i usually hate ALL turn-based games without exception and with passion!

Chrono Trigger is also great and considered by many one of the best, if not actually, THE best RPG of all time.

Of course Legend Of Zelda, furthermore Secret Of Mana, Secret Of Evermore, Terranigma - these are absolutely no-brainers.