There are a bunch of Final Fantasy games on the iPhone, including 3 or 4 only on the iPhone.
If you're into shmups you can't get better than Cave games. They are worth the premium price. I suggest Bug Princess 2, but they are all good.
Great Big War Game is my turn based strategy game of choice at the moment. It has multiplayer, I haven't tried it but it's there.
Carcassonne, based off the board-game, has been on my phone since the initial download over a year ago. Catan has an accurate iPhone adaptation but I do not enjoy the tempo of the game itself.
There is also Doodle Jump which is very good and very simple. More of an endurance challenge.
The Act is an excellent experience. Not too much longevity, especially for being more than a dollar. Much better than the Dragon Lair games for the iPhone, which I consider to be unplayable.
Plants Vs. Zombies, if you like the PC version you'll love the touch controls. PopCap periodically updates it with significant patches and it seems they will continue to do so.
Ascension is fun as well. You can really only play against the AI at this point as there doesn't really seem to be anyone playing it online. I played that one for a bit and got tired with it after a week or two.
I would be up for a game sometime.
Othello