FFVI, with a nod to FFX.

FFVI has:
Locke and Rachel's story~
Locke is in luv with Rachel and wants to marry her. He takes her into the mountains to find her an engagement ring. A rickety bridge begins to collapse under Locke, so Rachel pushes him out of the way and falls instead. She doesn't die, but she loses her memory. The people of her hometown blame it all on Locke, so he goes away. While he's away, Rachel is killed in a raid by the empire. In her final moments, her memory returns and she sends word for Locke to come back. He does so, to find her cold dead body. But Locke can't let go, so he has her body "preserved" in the hopes of reviving her one day. After the world is destroyed, Locke finds a way to revive his Rachel through the life-giving power of the Phoenix, but oh wait the magicite is cracked and Rachel is revived only long enough to tell Locke to move on. And then she dies.

Cyan's family's story
Cyan's wife and little boy, his king, and almost all the people of his kingdom, are poisoned and killed by Kefka while Cyan's away trying to kill the commander of the empire's forces. Cyan gets to find all their dead bodies. He deals with survivor's guilt for the majority of the rest of the game. Oh and did I mention he gets to see them get on the train to hell, and say goodbye AGAIN?

Cyan and Lola
When the world falls apart, Cyan finds himself alone on top of Mt. Zozo. He also happens upon the letters of a poor miss Lola, whose boyfriend was an imperial soldier (who you meet earlier in the game. He's injured and can't write, so you can send letters to Lola in his stead) and who has stopped writing to her. What she doesn't know is that her boyfriend died. Cyan, twisted by his own guilt and fear, writes to poor Lola as if he were her boyfriend, leading the poor girl on. Eventually, upon returning to Lola's home in Maranda, Cyan can do the right thing and...leave Lola another letter that explains he's been lying to her and her boyfriend is actually dead. Good times, yeah?

Did I mention the part where at least half of the entire world is destroyed in a massive cataclysm?

Terra and Mobliz
Never mind the story of her family, her heritage, the destruction of the Espers. Never mind the fact that she was controlled by the empire and forced to kill people. Terra's story in the world of ruin, the post-cataclysm version of the world, is that she finds an entire village of orphans. Kefka kills 100% of Mobliz's adults. Kiddies get to watch their parents die, then fend for themselves for however long they've been alone before Terra finds them. And Mobliz is just a microcosm of the entire FFVI world, since nearly the whole world was gutted in one way or another.

Gau
How about Gau, whose father thinks he's a monster and sends him away to live on his own like a wild animal? Seriously, they're cute little sprites so it's not as impactful, but this is some dark trout.

General Leo
A light in the darkness. A rose among thorns. The few good men of the empire all fall to tragedy. The best of them all, General Leo is brought down way too soon. Mainly because I wanted him to be a smurfing playable character dammit. So cool. So depressing.

Setzer and Darill
Another tragic story of love cut short. Darill was Setzer's lover and fellow airship aficionado. Setzer challenges her to a race, and when he can't keep up she flies away into the distance and is never seen again. Setzer finds her smashed ship a year later though. She flew too high, too fast. Lived too hard. Died too hard. Setzer takes her ship and buries it in a giant beautiful catacomb shrine.

Mog
He's the only one of his kind left in the world. All his people are dead.
RIP Mogsy, Mogwin, Mugmug, Cosmog, Moggrie, Moglin, Mogret, Moghan, Molulu, and Moguel.

Celes
Yeah, Celes. Just everything. Cid, the empire, her unrequited love of Locke, her lame special skill.

And there's more, but I hope you get the picture. That should cover it, yeah? FFVI, hands down.