Skies Of Arcadia. It's the anti Final Fantasy because it's cheery. The story is full of twists and turns and lovable characters who actually change and grow in the story. It does all of this without being an angst fest like many Final Fantasy games.
Eternal Darkness: Sanity's Requiem AND Odin Sphere, both of these games are perfect examples of how to write a multi character "Rashomon" style plot. Both games take a varied cast of characters and juxtapose them against each other to tell amazing stories. They're both very differend games, with Eternal Darkness being a mindscrew survival horror and Odin Sphere being a How do we save the world from disaster action RPG. But they're strength is their stories. Plus, they're fun to play.
The Legacy of Kain series is awesome. They had 5 games, multiple cliffhangers from executive meddling, and a plot thicker than pea soup if you didn't have a guide or a very good memory. However, they made that into a cohesive whole that ENDS with Legacy of Kain: Blood Omen 2. Despite the fact that Blood Omen 2 is the weakest of the series, it IS the sequel to Defiance.
Okami I'll second, as well as Bioshock. Metal Gear Solid 1 and 3 are the best stories to me. MGS 2 kinda falls apart at the end, getting a little too postmodern (My response to the end of MGS 2 was to say What the BLEEP!? Snake are you feeling alright? Let me check for a fever.) MGS 4 had a good story, it just needed an editor to cut about 10 hours of the cut scenes out. Minimum.
Shin Megami Tensei: Devil Survivor should be used as the principal example of how to make a silent protagonist game. On top of having a great and engaging story, you feel like you ARE the main character, instead of just controlling some green clad kid who can't say no to a favor. Not to mention that your choices affect the endgame in very significant and palpable ways by the middle of the game and determine which ending you get. (BTW: Honorable mention for Twilight Princess, it's story isn't one of my favorites, but it's still good.)
The World Ends With You has a great plot too. "How to use amnesia without it being cliched" "How to develop a character." "How to keep the player guessing until the final reveal, upon which they will scream WHAT THE HELL too. (In a good way.)"
I can't really choose one of these games to be the best of all of them. They're all fantastic in their own way and I recommend them all.
Hariman
PS: Final Fantasy X had a good story in there, I'll give it that. Unfortunately, it was buried under bad dialog, (The laughter, *Shudders*) idiotic characters ("Final Summon" used in a sentence like that would have made me ask: "Will this kill you?") with names that made me want to smack the localization team (Fozzy says: Wakka Wakka, so does Pac Man.), characters design from the cliche ridden shallow end of the story pool (Lulu probably takes a half an hour to pee, or she flips the dress over her head.) and not so random random encounters.
PPS: Dragon Age: Origins is chock full of good story too.
PPPS: Sorry for the wall of text. If you made it here, Skies of Arcadia, Eternal Darkness and Legacy of Kain (Full Series) are my top picks. Still can't choose a single one though.