A few of my favourites, excluding Final Fantasy:
Tales of Symphonia: it starts out very cliched but once you progress the story gets more and more interesting. There is a lot of character development and a very deep action based battle system.
Chrono Trigger: one of the greatest JRPG's ever made. Everyone who even slightly likes the genre should have played it, and thanks to the DS remake we poor Europeans finally can.
Paper Mario/Mario & Luigi: Absolutely the greatest Mario games besides the platformers, maybe even better. The games are very funny, full of references to the series and have a simple yet fun battle system. Even if you don't like the genre at all you will certainly love them.
Earthbound/Mother 3: Playing these postmodern games is an experience you will never forget.