The most important thing is the story, I don't like the usual story of save the world because they told us to do so.
Second, the characters have to be the less plain and shallow as possible (That's why I really liked FF13, the characters are top-notch)
Third, the battle system, I love turn based but also real time IF is fast, that's why I hate FF12 XD

The sense of self-improvement, how the story and characters grow on me, the sense of accomplishment after dealing with a challenge thanks to a correct strategy (management of resources).

Leveling up and music. Then Story comes after that.

I also don't mind grinding in a RPG, since I can sit there for hours and do the same thing over and over (hey, I like Dynasty Warriors games, so that says something :P). The only problem is that the grinding has to be fun, which is something that the Persona games (grind, try and create more Personae) and Final Fantasy XII had, IMO.

Exploration, quests, leveling, stat tweaking, finding items.

Story (along with characters), which is why I hate so many new JRPG's.

I think story is the most important but character and music help the story progress and set the mood of whats going on.

This.

Story then character romance and humours.

I'm surprised there are a lot of people saying story, considering the fact that most stories in RPGs are cliched...