User blog comment:Ikosarakt1/Uncountable function cannot exists/@comment-1605058-20130707061148/@comment-5529393-20130708174838

As LittlePeng9 said, there can be infinitely many ordinals of a given complexity, so there won't necessarily be a largest one. Also, you didn't define what the order type of a given function is;  rather, you defined the fundamental sequence of a given ordinal based on a given function.

It is certainly true that the fast-growing hierarchy's three rules are for ordinals with order type 0, 1, or omega;  So there is no rule for omega_1. Thus, if your question is, does the FGH define what F_omega_1(n) is, the answer is no. But, if you ask, can we choose a function F_omega_1(n) that eventually dominates F_a(n) for every countable a, the answer is, maybe. If the bounding number is larger than omega_1, then the answer is yes, we can always choose F_omega_1. If the bounding number is omega_1, then there exists a hierarchy f_a(n) for countable ordinals a such that we cannot pick f_omega_1;  however, there likely still exists hierarchies f_a such that we can pick f_omega_1. So the answer is quite up in the air.