User blog comment:Edwin Shade/An Injection From The Countable Ordinals To The Reals/@comment-1605058-20171219180546/@comment-5529393-20171219185454

This is where I make my obligatory note that you can insure that the slow/fast-growing hierarchies are strictly increasing under eventual domination by modifying the limit rule:

$$f_\alpha(n) = \max_{m \le n} f_\alpha[m](n)$$

So this can technically work if you use the axiom of choice to define all the fundamental sequences. The question is, is this any better than just using the axiom of choice to pick a new real at every ordinal? With that method, you can go all the way up to at least the continuum.