I have only had my Droid X for a few weeks so still have a lot more apps to explore. So far though, the non-game apps that I use most often are Audible and DC Metro Transit Info.
Audible allows one to access and listen the audio books purchased from one's Audible account. It has good rewind and fast forward controls. The only thing I don't like is that the elapsed playing time and time to the end is for the entire book, rather than the chapter. It would be nice to know how far one is in the current chapter. Also, you can't go online and purchase books with this app.
The DC Metro app is of course only useful if you use the Metro transit system in DC. It tells you when the next train is coming, and is generally as accurate, if not more accurate, than the station signs, which one can't always see from all parts of the platform.
My favorite built-in app is Swype. I think it's quite amazing. I don't know how it can tell what I am trying to type!