User blog comment:Pellucidar12/Idea for a function (Hyperprime counting functions)/@comment-30754445-20170723221034/@comment-1605058-20170723222207

Just because the sequence starts of slow doesn't mean it won't become fast. It is a very nontrivial question whether HP function is even well-defined (i.e. whether there are infinitely many hyperprimes), let alone finding its rate of growth (and showing it's not fast).