Dozens of things:
If I stick PDFs on it, I can view it on all my computers and devices. This holds for both contracts and ebooks.
If I stick pictures in the public directory, I can quickly show them to people
If I stick zip files in the public directory, it's faster than uploading to an SSH site
If do photoshop/fireworks/balsamiq mockups work out of dropbox, it's on all the computers, and can be pulled up on an iPad in a customer location.
If I stick the scanner on one computer with the output directory in dropbox, I can connect the scanner to one of the desktops, then have it automagically copy over to my mac laptop when done.