Kind of goes without saying, but almost anyone who does anything online these days needs access to Facebook. The official app is pretty robust, although it does come with some annoyances.
I actually kind of dig the official Twitter app (I was a fan of Tweetie 2, which is what this is). Does a lot of what you need out of a Twitter app, and for most users, it will be more than enough.
Angry Birds
Obvs.
Download
Adobe Photoshop Express
There are a lot of apps for improving the photos taken by your iPhone. I like PS Express for simple adjustments, such as brightness or contrast.
Download
MapQuest: Free Navigation, GPS, Maps & Traffic
Free turn-by-turn directions? Don’t mind if I do!
Kindle – Read Books, eBooks, Magazines, Newspapers & Textbooks
Maybe not essential if you don’t actually buy digital books from Amazon, but the e-book reader for the Kindle is amazing (and free)!