"Add More Values" doesn't work during user registration

I currently have some multigroup fields shown during registration but but when a new user clicks the "Add More Values" button there is a pause and then a new blank, draggable box appears without any fields. I've tried giving anaonymous users various permissions (some that I wouldn't want to leave on) but nothing seems to work. I do have the permission enabled for "create profile content" (profile is the name of the content type). I'm also using the Content Access module because some of the info entered during registration is not going to be publicly viewable.

It works great when I'm logged in and editing an existing profile. I click "Add More Values" and the new set of fields shows up and will store just fine.