There are tons of fun and useful apps. My faves include:

The google app -- you can talk into it and it will go out on the interwebs and get the answer to your question.

Shazam -- you can listen to a song and it will go on the interwebs and figure out what it is.

Google earth -- the iPhone version of the desktop app. Fun for time killing in the dentist office.

SnapTell -- take a picture of a book cover, CD or DVD and it goes out on the interwebs and can tell you how much you can buy if for online (or more information). Unlike the Amazon app, it only does books/cd/dvd but it is quick and doesn't require a human to figure it out.

Pandora -- internet radio (also AOL radio).

Yelp -- nice front-end for yelp.com

UrbanSpoon -- helps you pick a place to eat

Darkside is a nice flickr viewer, but flickr.com now is optimized for iPhone so I use it a lot.

If you use twitter, get the free version of twitterific.

Sites like tuaw.com will and iphoneatlas.com will have reviews of apps.

Lofi.metachat.org is formatted for easy iPhone usage

I've been using Lose it!, a diet tool/calorie tracker and (Xmas not withstanding) it's awesome for keeping me on track on my diet

I like Xpenseit which can track income and expenses and is hell easy to use, WorkLog to track time, klick to access flickr, weightbot to keep on eye on things there, To Dos for task lists and iBlogger to post to a few blogs (although I'm looking forward to this being tweeked a lot more).

Jott looks like one of those things that will be really amazing once I take the time to figure out exactly how to make the best use of it. (Story of my life: get gadget that has vast potential to get me all organized and efficient; then completely fail to make use of all that potential on account of how I got no time to figure it out, on account of how I am disorganized and inefficient.)

I use Instapaper a lot; it's a very easy way to save stuff that I come across on-line and want to read later when I'm away from my desktop, like when I'm standing in line at the DMV or waiting for the dentist or something.

Evernote is awesome and integrates beautifully with the iPhone, thereby allowing me to dispense wholly with the iPhone's crappy native Notes app.

Oh, and I love the Amazon app, not for doing ordering, but for checking reviews when I'm out shopping and am tempted to buy something. ("Hmm...two stars? Drat." *returns shiny tempting object to shelf*)

Shazam -- you can listen to a song and it will go on the interwebs and figure out what it is.

Heh... I just used this (on my G1 phone) to figure out what that familiar song was, playing on the boombox next to the cash register in our cafeteria. It was Paul Hardcastle. No wonder I couldn't come up with his name... I haven't thought of that guy in 20+ years.

The Weather Channel app. Way better than the built-in weather thingy.
some wikipedia reader (i use wikiamo).
facebook
some twitter (i use twiterrific)
flashlight (yeah, i used it once)
level (never used, but i know someday it'll be justified)
itunes remote if you use itunes
I personally use IOweYou.
Translator
Dictionary
Taxi (gives you taxi phone numbers)
Free games: Aurora Feint (Beginning), JellyCar, Trace, Cube Runner, iBall3d
WiFinder (scans all wifi visible for free ones (checks for auth pages to see if they're really free))
Stanza (ebooks)
UrbanSpoon (The slot machine for "where should I eat" solves the basic problem of mankind)
Some Movies app (i use NowPlaying)

Around Me and the iTunes remote are about the best, I think. I also use an app called Tides that gives you tidal predictions for wherever you happen to be (not much use to most, though, I guess).

I noticed the other day that I use it for all sorts of things, almost all of which are not phone-related.