Here's my list of commonly used, non-standard apps for my Droid X:

WidgetLocker - A great lock screen replacement.

Tasker - A Swiss Army Knife like tool to do things such as custom buttons, alarms, car location tracking, automatic actions when connecting to particular Bluetooth devices (example: start Pandora, set the volume to medium and play when connecting to my stereo BT headset).

Pandora - The popular custom radio app.

IMDb - The Internet Movie Database, better than using a web broswer to search for movie info.

WikiDroid - like IMDb but for Wikipedia.

FreeDictionary - I haven't found a great dictionary app but this one is okay.

Google Translate - for translating

Google Voice - for transcribing voice mail to text and for international calls.

TeslaCoil LED - a flash light using the camera's LED.

And a novelty one:

Instant Heart Rate Monitor - Uses the LED and camera to measure your pulse--clever!