Write for Dropbox is now available for iPad as well as iPhone (where it's now free-of-charge), and certainly looks like a nice solution for word processing straight to your synchronised file storage.
I've been heavily reliant on Dropbox for a few years, and it's increasingly becoming the back-end file storage solution for a whole range of apps that I use daily. It also provides much of the glue between my mobile workflow and what I need to do when I get back to the desktop (there are still a few things I like–or need–to do sat at my big-screen iMac). I use iCloud too, but I treat that in a rather different way–for now.
If I were building an app right now that needed reliable file syncing across different platforms and devices I'd most likely choose to build it on Dropbox, though a bit of me would still be nervous about trusting a critical part of the app to a third party. If you don't think that matters, you should check out Brent Simmons' excellent article from a few weeks back.
Update: Jamie Bullock is absolutely right when in that plenty of writing apps already use Dropbox for file syncing, and that it's surprising how long it's taken Dropbox to create its own solution. What's significant though is that Dropbox appears to be recognising that it's built a platform, and starting to build its own ecosystem of tools on top of it.