The more precise statement of this question is partly inspired by the question Extensions of fast-growing hierarchy. However, I didn't want to derail the original question (since the OP may have completely different set of questions in mind), so I am posting this as a separate question.
Suppose we are given some fixed family of fundamental sequences for limit ordinals up till $\omega{_C}{_K}$, but not necessarily including $\omega{_C}{_K}$ itself. Now suppose the family of sequences is described by the function $F:\omega{_C}{_K}\times\omega \rightarrow \omega{_C}{_K}$. The obvious meaning of $F(\alpha,n)=\beta$ is that the n-th term of the fundamental sequence for $\alpha$ (whenever $\alpha$ is a limit) is $\beta$. When $\alpha$ is not a limit we may take $F(\alpha,n)=0$ as convention for example.
I use the same definition of notation function $address:\alpha \rightarrow \Bbb{N}$ as in my other threads (not mentioning it here to avoid excessive repetition): Finite Number of Registers and Computable Well-Orderings. It is just below the bold heading First Part.
Now for any limit ordinal $\alpha$ define the function $F_\alpha:\alpha\times\omega \rightarrow \alpha$ (sorry I forgot to add that the function $F_\alpha$ is suitably defined unique "restriction" of the function $F:\omega{_C}{_K}\times\omega \rightarrow \omega{_C}{_K}$). That is $F_\alpha(\beta,n)=F(\beta,n)$ for all combinations of $\beta$ and $n$ in domain of $F_\alpha$. Given any computable well-ordering (on $\Bbb{N}$) of order-type $\alpha$ ..... with corresponding address function as $address:\alpha \rightarrow \Bbb{N}$ ..... we define the representation function $f_\alpha:\Bbb{N}^2 \rightarrow \Bbb{N}$ of $F_\alpha$ as: $$f_\alpha(address(x),address(y))=address(F_\alpha(x,y)) \quad for\,all\,\,x\in\alpha,\,y\in\omega$$
We can say that the function $F$ is step-recursive (sorry that's not the best term ... I can't think of a better one) iff for arbitrarily large limit ordinals $\alpha<\omega{_C}{_K}$ there exists atleast one recursive well-ordering (on $\Bbb{N}$) of order-type $\alpha$ such that the representation function of $F_\alpha$ is recursive in the given well-ordering.
Now suppose we define some specific hierarchy (say fast growing hierarchy) corresponding to the function $F$. Denote the function that is formed as a result of the hierarchy at level $\alpha$ as $g_\alpha: \Bbb{N} \rightarrow \Bbb{N}$. Define the collection of function $G$ as: $$G=\{\,g_\alpha\,|\,\alpha<\omega{_C}{_K}\,\}$$
Now suppose that the function $F$ is step-recursive. We have the following two questions:
(1) Does there exist a function $F$ such that the functions contained in $G$ are all recursive and there is no total recursive function that dominates all the functions in $G$?
(2) Does there exist a function $F$ such that the functions contained in $G$ are all recursive and there exists a total recursive function that dominates all the functions in $G$?
Also how sensitive would the answer to these two questions be to specific definition of hierarchy (say replace fast-growing with slow-growing). Any kind of references will also be quite welcome.
Few Extra Questions
Can we describe somewhat more general techniques that can help us to answer the previous two questions above, given some $F$ that is step-recursive (as opposed to finding only one example of such an $F$). Do the two problems above becomes trivial if we drop the requirement for $F$ to be step-recursive (and how)?
P.S. I don't have any personal experience with hierarchies at all. However, this question has been on my mind for some time (thought the precise description wasn't). Also if the proof-theory tag isn't quite appropriate for this question then it can be removed.
