For quirky fun, play Brave Fencer Musashi or Threads of Fate (Dew Prism in Japan) They are both pretty light hearted but really fun.

For a more modern twist try out Parasite Eve, though it is a bit gory and weird...

Front Mission III and IV are damn fun to play if you like SRPG's that involve using giant robots and heavy political themes.

Einhander is good, it's just trying to find a copy that's difficult. It's a pretty good shooter only made impressive by the fact it was made a company that doesn't really specialize in them.

The Saga series is unique, it is sorta've Squenix's unoffical testing series. All the games are pretty experimental. Some good (Saga Frontier 2) some bad (Unlimited Saga)

Chrono Trigger is a classic game which revolves around time travel and sword and sorcery. It has incredible graphics and music for a game released late in SNES life. It's sequel is also pretty good, though it has some mix feelings in the fan base. I suggest you try it out for yourself before listening to anyway really give their opinion about it.

The Mana series is a a personal favorite, if you can get a copy of Seiken Denetsu II (Secret of Mana) for the SNES I would go for it. It's a damn good game and will prepare you for all the others in the series. Legend of Mana is pretty good though it is pretty aimless at times and doesn't really have an overall plot. It is still very pretty and really fun to play.

I save my two favorite non-FF's for last. Xenogears and Vagrant Story. Xenogears, is an incredible game with an intricate twisting plot and a storyline that just blows you away. It uses a few RPG cliche's but it does them in a good way. It has one of the greatest soundtracks in an RPG and a wonderful cast of characters that you grow to really care about. The battle system is really fun but even I'll admit that towards the end of the game, it gets a little boring but the game is LOOOOONG expect to put at least 60+ hours into it.

Vagrant Story is my other favorite. Made by the FFT team, it is also set in Ivalice (though they never blatanly say it, and it's odd having a real religion showing up in a ficticious world) It's battle system is really weird and gets some getting use to, especially since the game doesn't have a real tutorial. Once you get a handle on combat, making weapons and managing Risk the game is really fun. In reality, it's the story that is the true highlight of this game. It's like a medieval Metal Gear Solid, involving a Riskbreaker (a feild agent rank within a secret task force created to cover up the secrets of nobles and high ranking officials) named Ashley Riot being sent into a cursed city to rescue a Duke's son who has been kidnapped by a cult that was being secretly funded by said Duke. Ashley must rescue the child and eliminate the leader of the group, a man named Sydney Lasserotte, before the Crimson Blades ( A fanatical group of knights under the authority of the Christian church) get to them first. It's plot is incredibly well written and is one of the reason's why I've been looking forward to FFXII since I heard this team was developing it.

On the Enix side, I suggest Drakengard, another game with a really freaky plot... or Star Ocean: The Second Story or even Star Ocean III: Till the End of Time. SOII has a wonderful cast and though it's plot is kinda've weak it's still incredibly fun to play. Word of advice though, choose Claude if you want the better plot, choose Rena for the better party options. The game also has the best way of going about doing optional dungeons. As you know, 9 at of 10 times, most RPG's have uber optional bosses that require insane amounts of time to prepare and beat. Afterwards, the final boss is a joke since you beat this uber boss. In SOII, to unlock the optional dungeon you have to do a side quest that makes the end boss insaely more powerful than normal. SOIII has a better storyline but I feel the cast isn't as solid. It also has the most bizarre plot twist in any game that tends to either make or break the game for people (I still know people who feel this game is dead to them since they reached that plot twist)

If you can find it, Robotrek is another wonderful Enix game that features robots and time travel but it is for the SNES