There's quite a few I use, both regular and jailbroken apps.
For the regular ones:
The Google app is a good one if you use any of the Google Services. Lets you use your voice to do a search for something.
Shazam is a great app for identifying music that's playing. There's a free and a paid version of the app.
Evernote is a great note-syncing app. I use it for quick access to equipment manuals (PDFs) for work and for quick notes. They can sync on any computer that you run Evernote on.
Meebo - great multi-protocol IM program. Yahoo, AIM, MSN, Google, etc. Even does push notifications when the app is closed.
Wunderadio - Nice internet radio app. Also does regular radio stations that have internet feeds. Uses Safari to play in the background.
For jailbroken apps:
Intelliscreen - turns your lock screen into a nice little daily planner - it has a highly-configurable layout and can show you RSS feeds, Weather, appointments, email and such. I use it for weather and for my calendar...that way I only have to push one button to check.
BiteSMS/iRealSMS - both are nice SMS replacement apps for Apple's default SMS app. Gives you lots of other options including quick send/reply and group messaging.
ToneFX - allows you to have different sounds assigned to system events, such as replacing your incoming SMS notification. They have a desktop program you can use to make your own sounds.
Shrink - allows you to custom-resize your app icons. Can be used in conjunction with other apps to allow more icons per screen. I used this to give me 25 apps per screen before I upgraded to iOS4.
sbSettings - a nice little program that gives you more control over your phone. You swipe the bar at the top of the screen and a drop-down screen lets you turn on/off different elements of the phone such as data, 3g, wifi or turn the phone radio off (airplane mode). You can also dock apps to this to put handy apps a few taps away.
Anyways, that's just a few of the apps I have and recommend.
