My apps, of course.
I have a Motorola Atrix, which I bought for £250 in July. I pay £10 per month for "unlimited" internet, unlimited texts and 250 minutes of calls. I'm thinking of switching to another provider, though, who provide unlimited internet without the quotes around them for £15.
Seriously, I use the Economister app I made literally every day... which is why I made it...
Apps which I don't believe anyone has mentioned but are still useful include:
LUKS Manager - On the fly disk encryption. Very cool, easy to use interface, but requires root and probably a different kernel. There are more accessible disk encryption utilities, but this one lets me start and stop things with scripts and such. Anyway, I just wanted to trigger in people's heads that you can encrypt your personal files on your phone, in case you lose it.
BetterBatteryStats - Tells you which apps are draining power (free from XDA forums)
Swype Beta - My favourite keyboard. Beta version is much better than the Swype that comes with your phone.
WiFi Keyboard - This lets you use your computer's keyboard to type onto your phone via WiFi. Useful for typing out long emails on your phone (I can access my work email on my phone but not my home computer), for playing games, typing in terminal commands, etc.
GScript Lite - Useful to run regularly used terminal commands on your phone (can create shortcuts on homescreen)
DiskUsage - Visualises your SD-card to see what's taking up the most space on it.
QuickPic - Replaces your Camera Gallery. Much better than Google's, probably better than the one that comes with your phone.
KeePassDroid - stores all your passwords, works with the desktop version
Samba Filesharing - Shares your phone's SD card over wireless LAN. So on Windows you can map your phone to a network drive (mine is Z: and Y: ) so that you can store files on your phone and seamlessly use them on your computer. Your computer just sees the files at e.g. "Z:\personal\Dream Journal.docx". It means that (a) you don't have to plug your phone in via USB to transfer files, and (b) you don't need to worry about syncing two different versions of your files - you just keep one version on your phone, and your computer thinks it's just stored on a network drive.
Instant Heart Rate - You put your finger over the camera lens and it tells you your heart rate. It does this by using the flash on your phone to light up your finger, and then the camera detects changes in light intensity as the blood passes through your fingers.
WiFi QR Generator - VERY handy when your friends come round and they want your WiFi password. It generates a QR code for your wifi, which includes the password and allows your friend to just scan the barcode in the normal Barcode Scanner app and connect automatically.
EDIT: Remembered another one:
Smart Tools - can measure angles, sizes, distances, and so on. Not accurate enough to depend on but handy in a pinch.