I have an iPhone 4 and consider myself a heavy user of apps.
For social networking
LinkedIn
Facebook (family only stuff)
Foursquare (location based app - it's fun)
TwitBird Pro (you can follow me on Twitter @affygility)
For business
Salesforce CRM app -have to keep in touch with those valuable clients
i41CX+ calculator. It's the best, mimics the HP 41CX RPN calculator. For the nerd in the family
Evernote. Syncs lots of data between the desktop and the phone
Dictionary - can't spell it will find out how to
Analyticsapp.com Gives me real-time google analytics information from our website(s)
Dropbox - file sharing app between iPhone and desktop
NIOSH Pocket Guide for Chemical Hazards - need to know an occupational exposure limit...go here.
For music
iPod
Pandora
Shazam - always dead-on at recognizing the song.
For news
CNN
SlingPlayer - allows me to watch my home tv, no matter where I am. It rocks!
Sports
Ski Report - I live in Colorado, need to see if
Breck has some fresh pow!
ESPN Sportscenter
Travel
Kayak
Yelp - need to know where the best places to eat?
The Weather Channel
Avis rent a car
I travel a lot, so many of the apps make it easier for me while being a road warrior. Everyone in the family has a smart phone, so it's easy to stay connected.
This is just the main ones, but doesn't include games, individual site apps, etc. Have fun!