Dropbox allows you to Open In.. files that you have received in your email. This prompts you to select a location in your Dropbox folders to upload the file to.
So if an audio recording app for example, doesn't support direct upload to Dropbox but does support email, email the file to yourself and once that email is received, tap and hold to see the Open In... options.
Select Dropbox from the list and the Dropbox app will open and request the upload location. You can even rename the file at this point if you wish.
So far, I haven't found any file types that don't upload but I'll comment on this post if I find any.
I would love to know if the android version of Dropbox supports this too. Please comment if you know! :)