I have a Verizon Motorola Droid, but I'm sure that most of the same apps are available for iPhone.
Some of my faves (all free):
Accu Weather and The Weather Channel (similar, but I'm still figuring out which one I like best)
Advanced Task Killer Free (closes down unused applications to help keep your battery from draining faster)
Backgrounds (great wallpapers)
Barcode Scanner and Shop Savvy (very similar, but I'm still trying to figure out which one I like best...these are great for comparison shopping)
Calorie Counter (has restaurant calories too)
Cat Sounds (this freaks my cats out, lol)
Dilbert (comics!)
Healthy Recipes (by SparkRecipes.com)
Facebook
i Music and Imeem Mobile (search for specific songs, listen for free, download for free)
Movies (reviews, showtimes, etc)
My Coupons
Nubi News (good news portal)
Shazam (can identify any song playing, gives you title, artist, etc)
Slacker Radio (similar to Pandora Radio, but I like this one better)
Tip Calculator
TV.com (I don't think that works on the iPhone though?)
Useless Facts
Where (works with the GPS built into the phone, will locate different places, give directions, etc)
Wiki Mobile (Wikipedia)
YZ Free (Yahtzee game...I'm addicted to it, lol)
I recommend browsing the App Store, and looking at the most popular free apps. Download what looks interesting, try it out, and then choose to keep it or delete it. There are apps that you pay for too, but I haven't had the need for any yet.