I am completely addicted to my iphone. It has solved any issue I ever had about forgetting my phone.
Some of my favorite apps:
Twittelator Pro - I am also addicted to twitter and this is by far my favorite twitter app, although Tweetdeck is a good free option, just doesn't do as much.
Pandora
Evernote
Alarm Lite - I use this so that I have a visible clock while lecturing and as an alarm clock when traveling (although alarm is not very loud). Must be plugged in for overnight use.
TextFree Lite - I don't text very often and usually only to one or two people, so I don't have a texting plan. With this app all texts that you send are free, and if your friend also has the app and uses it, the texts that you receive from them will also be free. Saving me lots of money this summer as our friends are in Germany and the kids are using it to keep in touch.
Read it Later - I use this on both pc & iphone to save links (articles, blog posts etc) so I don't forget to go back & read them.
Amazon Kindle - turns iphone into kindle.
Flight Status - not free, but very handy if you travel a lot or if you make frequent airport runs to pick up others.
Wordpress - to keep blog updated. I haven't actually used this yet - I don't blog often and I usually want to add lots of pics.
Epicurious - cool recipe app - will generate a shopping list for you.
Pocket Informant Lite - great app that has calendar and to do list together in one place.
PayPal - Just found this (Thanks Lew!!) Can now easily use PayPal at shows.
Google Earth - just plain cool
Edit: Just added Stanza - looks very cool!