Which iOS app will let me paste an image? TL;DR; Version:
I'm looking for an app (preferably free) that will allow me to past an image from the clipboard and save it either to my photo library or to an online service.

Long Version:
Several months ago I had some photos stored in my Dropbox which accidentically got deleted without my knowing.  I recently realized the photos were gone and that I had no backups of the photos.  The other day I was in the Dropbox app on my iPad in an area which did not have Internet access.  I noticed that Dropbox had in its cache a bunch of the photos I had lost.  However, Dropbox will not allow me to import the photos into the photo libary or print them or share them; all of these actions return with the error "Unable to Connect to Dropbox - There may be a problem with your iPad's Internet connection."  I can't enable the internet connection because that would sync my Dropbox and loose the photos.  The "Copy Image to Clipboard" action does work, however I can't find any app that will let me paste the image and save it.

I have tried using Evernote, and I can paste the image, but the size limit for notes prevents me from syncing the note.  I've tried GMail and I can paste the image but when I send the message the received e-mail does not have the image attached or included in any way.

Is there an app out there that will let me paste an image and either save it to the photo library or sync with an online service?
Since you're able to copy the image, open up the Messages-app, open a new message to your own iMessage-address, and paste, then send. You should be able to save the image to the camera roll from there.

On newer iOS like 10 and higher, the ubiquitous share sheet function lets you move documents around so much easier, that third party apps aren't needed for this basic function anymore.