This, still hasn't been matched

Most of the Mario RPG games have a great story, especially SMRPG, Thousand Year Door and Bowser's Inside Story. Considering they're the kiddiest RPGs out there, it's surprising how they manage to involve the player of seemingly any age.

Xenogears, Xenosaga I, Xenosaga III, Xenoblade Chronicles, Chrono Trigger.

Not even the best stories in their series. 

Xeno-series as a whole
Persona 2
SMT: Nocturne
Paper Mario TTYD
Final Fantasy VI
Final Fantasy VII

Skies of Arcadia... Now that's a good one.

I was thinking on this, and I found it kind of hard to answer. I like a lot of JRPGs, but I wasn't sure if it was because of other elements, such as combat system, characters, music, etc. When I think about JRPGs I love on story alone, I realize the stories aren't all that interesting.

Perhaps Chrono Trigger and Tales of Symphonia?

Xenogears is in a tier of its own when it comes to story. It is to date still the best story in a videogame ever.

Lost Odyssey. The main story was good but it was the side stories that were excellent.

For the recent FF games, I think FFXII had the best story.

Shadow Hearts : Covenant, decent story with humor too. :( Most overlooked JRPG ever.

Fire Emblem: Path of Radiance is my absolute favorite RPG (and possibly my favorite video game), so it goes without saying that I loved the story. The way the game's war was intertwined with a young man's coming of age story, the fantastic lore, powerful themes, and how well written even minor NPCs were...everything about this game's story hit the right notes with me. The follow-up, Radiant Dawn, was a worthy sequel, but the first game still holds a higher spot in my list of RPGs.

SMT: Persona 4 and Devil Survivor 1 and 2. Admittedly the second one did get a bit convoluted, but I still really enjoyed it. It kept me interested enough to play 120+ hours anyway.

Tales: Symphonia and Abyss were both great, and had some strong themes (discrimination, value of life) that really hold up for me even today.

Xenoblade goes without saying...but I'm saying it anyway. The twist in the end blew me away.

Probably a lot more I'm missing, but these are the ones that come to mind right now.

I created an account just to say Xenogears. That game blew my mind.

There are many others too, of course. Many FF games have a great atmosphere. Suikoden games are pretty epic. But to be honest, many great RPG's have only a mediocre story, like Dragon Quest VIII, the Grandia games, or FFVIII.