I think I'll throw in my two cents. If you're looking for newer RPG's, Bravely Default could very well be a masterpiece, in my own personal opinion. Then there is just about every game in the Tales of series. Tales of the Abyss, Tales of Symphonia, Tales of Vesperia, Tales of Graces, Tales of Xilia... all of them are great. My personal favorite is Symphonia.
Back in the realm of the olden days, I see one game that is distinctly missing in all of these lists. Where is Earthbound? Not to mention Mother 3, on top of that. Simply incredible games, in my own unprofessional opinion. Chrono Trigger has already been recommended to you, but I must say, it's one of the best RPG's out there to this day. The story is incredibly compelling and most certainly will pull on your emotions like a beautiful love song. On top of that, the gameplay and fighting system is simply astounding. Lets also not forget out our sweet child Fire Emblem, either.
And to break from the caste of standard JRPG's, I would highly recommend Metro: 2033 and Last Night. They may not be what you are looking for, as they have about as much in common with Final Fantasy as Halloween and Halloween 3, but the story of the series is simply incredible. I'd also recommend the earlier entries of the Silent Hill series in this similar vein, particularly the first and second.