There's lots of apps that I like, but the ones that I use the most are:
1. Kindle for iPhone - it's it's my top pick for taking a book on a trip with me. Plenty of free books, and easier to use for reading than I thought it would be.
2. Keeper - an encryption app for storing passwords and such. I use it all the time.
3. Peggle - my favorite game for the iPhone.
4. GeoDefense - another very enjoyable game.
5. BeejiveIM - an IM app that's compatible with all of the usual ones, and has push notification.
6. Facebook - if you are at all into Facebook, this is a very good app.
7. Shazam - very good for identifying a song when you hear it on the radio.
8. AroundMe - great for finding restaurants, gas stations and ATM's when you need them.