Valkyria chronicles has one of the most engrossing stories. I also agree with the persona 4 votes.

Suikoden 2 probably but Breath of Fire 3 and Vandal Hearts come close. 

-Mother, Earthbound, Mother 3, the entire series has a fantastic story.The atmosphere is great, which complements it well also.
-Pokemon Mystery Dungeon R/B; D/T/S- First game that ever brought me close to tears. Fantastic stories that are unmatched in the franchise.

It's minimalistic but the the story in Nocturne is very interesting

I didn't play that many JRPG's but from what i've played, Persona 4, Xenoblade Chronicles and Most Fire Emblem's have great story. Especially Xenoblade's blew my mind a few times.

+++

Also Persona 3 as a superior evangelion.

Final Fantasy 6 deserves a crown for the category too.

100 % YES! i play ca. 50 rpgs in my life and PERSONA is so amazing with the story. Persona 3 too.

it is incredible! everyone should experience it.

The Xeno series

Outta the top of my head...Xenogears, Chrono cross and Nier have great plots. If you're averse to tought then something like Suikoden 2, Persona or Grandia is better for you.

Suikoden II, but the fist one is not bad either. Other personal favorites of mine are both Lunar: Silver Star Story and Lunar: Eternal Blue.
From recent JRPGs,the only ones that come close are The Legend of Heroes games.