Higurashi - When They Cry, being a horror/mystery visual novel, kind of HAS to have a decent story. Part 1 is free until August 15 if you want to try it out.
I'd agree with GTA, that's got a gripping story, which is why I came back to it day after day til it was complete.
I could actually imagine Fruit Ninja with a story...Basic outline: You are a poor guy out on the streets when you impress a ninja master with your fighting skill when you get attacked by wild strawberries. He takes you in, and...that's all I've come up with so far
I think hybrid had the story that sucked me in the most of any iPhone game. Ibreally cared about what was happening. To bad gameplay was just ok.
I really enjoyed the story of Chaos Rings, and as someone already said... you can't really beat Secret of Monkey Island when it comes to storyline ( Paper Toss come's in close though )