I'd say give Catherine a shot. The game's very well written in my opinion. The gameplay is smart and ties into the story nicely. It brings up a lot of interesting questions. Also it has a totally unique artistic style (among games, at least).
I just finished The Darkness for the first time, and that game has a killer story. The mechanics of it are relatively standard shooter fare, but there are some unique features that really drew me into the protagonist character and made me understand his internal conflicts.
Some pretty obvious suggestions: Bastion and From Dust. Bastion, in my opinion, suffers from having this kind of accessory hack-n-slash gameplay that doesn't always add much to the story, but it is a good experience anyway. From Dust has some frustrating implementation problems that keep it from being great, but I still loved it.
Have you played many indie games? Jason Rohrer did some pretty interesting stuff, I'd recommend Gravitation especially. Messhof's games are cool, Randy Balma: Municipal Abortionist was one of my favorites. Psychosomnium by Cactus was good. These are all extremely short, minimalist games, unfortunately. I'd love to see more recommendations for full-length, professional quality games.