I've recently finished two iPad games. That's actually a big deal (relatively-speaking).
First off, I don't play many games, as I don't have a lot of discretionary time, and if I did I'd rather write programs than use them.
Secondly, I'm particular about what games I play and I'm also not that good at most games. So if I don't really like it, or it's too hard, I just don't finish, though I have to say that once I'm really playing a game I stick with it, and intend to complete it.
And finally, I suspect that most iOs games are never finished. I'd love to see some numbers on this.
So I finished these 2 games, Machinarium and iGadget, and really enjoyed them. I've been thinking about why. Even though they were vey different from each other, they both put their story, their plot, at the center of the game. In fact they actually sacrificed game play (Machinarium through a built-in cheats system, iGadget by limiting game-play) for the story's sake. A little further and they would be an interactive book.
If you like the sound of that, I'd suggest these two games.