I'm queen of the free apps!

Here are some I recommend:
Arcade Bowling Lite: Skeeball app that's a lot of fun. The lite version doesn't allow you to save high scores, but the 99-cent version does.
Arcade Hoops Basketball: Another carnival game, this is a basketball shooting game.
BubbleWrap: Pop the bubbles for mindless fun! This is great to keep kids busy in the car, doctor's office, etc.
The Deep Pinball: Truly awesome pinball game. The graphics are great.
Dinosaurs: The American Museum of Natural History Collections: Excellent app. I love looking at the dinosaur picture mosaic and choosing random photos for further investigation.
Disney.com: Official app for all things Disney, including games.
Disney World Wait Times: If you're going to Disney World in Florida, this app will give you the wait times for rides in all four parks.
eBay Mobile: A must!
Google Mobile: Self-explanatory.
iGlowStick: Virtual glowsticks, which are seasonally updated.
Kindle for iPhone: Amazon Kindle app for reading books. After installed, you can go to Amazon and buy Kindle books. I found many that are free.
Mad Libs: Fun for kids. When mine start playing this, I have a hard time getting my iPod Touch back.
NASA App: Great for space fanatics.
PayPal: You never know when you might need to check on your PayPal account.
Rock Band Free and Tap Tap Revenge 3: Express your inner rock star -- I sure do!
Wolfgang's Concert Vault: The website has a huge assortment of recorded concerts that you can list to for free.
For book lovers, there are a lot of free apps for books, both new and classic. You'll have to search through those listings to find genres you like, but I've seen everything from romance to horror to Dickens.
If anyone is serious about genealogy, Ancestry.com has a brand new app. It will link with the info you've uploaded to the website, providing you with a mobile version of your family tree. After spending two hours walking through a cemetery over the summer, trying to find my great-great-grandfather's grave and instead finding a ton of info about other deceased relatives, I would have loved the convenience of this app.
There is also a website that lists all of the apps that are free for a limited time (usually one day only):
http://freeappalert.com/