Xenogears, Chrono Trigger, Riviera, Fire Emblem (7), Shadow of the Colossus, Zone of the Enders: The Fist of Mars and maybe Valkyrie Profile 2.
These are all good ones. Some might balk at the inclusion of Shadow of the Colossus, but I don't feel that bloated dailogue or redundant cutscenes equate with good stories. Shadow of the Colossus has very little talking, but conveys powerful emotions succinctly, which is something I always admire when I come across it.
A perfect example of a game that presents a story in completely the wrong way is Okami. Oh, sure, it presents the environments well, but the actual story is a muddled mess buried underneath mountains of completely irrelevant monologues from whiny support characters. Yahtzee talks about MGS4 needing an editor, but I doubt it's much worse than Okami. In the end, the story of Okami isn't even very deep, just referential of other, better stories.