Only the AJAX aspect (all fields) is failing. Navigating to normal profile edit works fine. I've no idea why AJAX is failing as it appears the following IF statement keeps failing for fieldclass usage (can't say for sure, ajax plugin is not reaching its storage at all so something on the way to it is failing).
Code:
if ( CBuser::getMyInstance()->authoriseView( 'profile', $user->id ) ) {
Was your install modified or anything of the sort? It's possible it's a permissions issue, but I don't know where specifically the issue is coming from, sorry. For me to continue you'll need to provide a clean install of just Joomla + CB + CB Core Fields Ajax as can't confirm any issue on any of my tests except on your install specifically.