I know you use imgur as a defacto img hosting site you'll display from, but imgbb is a rock solid site and doesn't have some of the downsides of imgur, such as, it's quicker (i.e. can paste images) and get links to various formats/media for the pic.
I'm seconding this as I can't make an imgur account (You need a smartphone, which I don't have) to put a image on my profile and I'd rather not upload a image to imgur without an account because I don't think I would be able to remove it later if need be. I know they have a removal request form, but I don't think that's a sure thing.
Have you tried using a landline? Many support receiving SMS messages.
Unfortunately our's doesn't. So that's out.
How about the other methods of signing up which don't require a smartphone?
