If you asked me about the first RPG that had what I would consider a profound effect on me, it would be Final Fantasy III for the Super NES (a.k.a. FF VI). To this day I hail it as the "best rpg ever." The number of characters and the amount of story that went into the game was incredible, and the graphics for the time were quite amazing...
I have never been able to pinpoint which was that FF I played on the SNES, because of all the troubles with those numbers. But now you quote it I remember now that was FF3 but in fact it was FF6. Everything was great but it was my first JRPG ever (never own any console and own a SNES only very late), and the music during fights, plus the fight system, plus all this extreme harassing of random fights, finally succeed make me upset again the game and I gave the console without to ever come back to the game.
Now I'm even older and succeed play more JRPG I'll be able enjoy it quite more, so it's cool a DS remake of FF6 is coming and that will be a great opportunity to come back to this game.