Does there exist a computable function that grows faster than fast growing hierarchy? Does there exist a computable function that grows faster than fast growing hierarchy for every computable ordinal $\alpha$?  Or does it follow that fast growing hierarchy grows as fast as any computable function?  I honestly can't figure out how to even understand how I could tackle such problems, and indeed I am fairly new to all this.  A quick definition of fast growing hierarchy for those who don't know:
$$f_\alpha(n)=\begin{cases}n+1&;\alpha=0\\\underbrace{f_{\alpha-1}(f_{\alpha-1}(\dots f_{\alpha-1}(n)\dots))}_{{n\text{ amount of }f's}}&;\alpha\text{ is a successor ordinal}\\f_{\alpha[n]}{n}&;\alpha\text{ is a limit ordinal}\end{cases}$$
where $n\in\mathbb N$ and $\alpha$ is some computable ordinal.
I imagine this is very hard to proof one way or the other, since the fast growing hierarchy can outgrow things like the TREE sequence, which are not provably total... so I honestly have no clue.
The most I can say is that FGH itself is computable.
 A: It's important to note that "$f_\alpha$" isn't really well-defined - if we pick different fundamental sequences, we get different functions.
Really, what you have when you get down to it is a function $f_n$ for each ordinal notation: for a given notation $n$ for a limit ordinal, there's a natural fundamental sequence associated to it, since we can enumerate the notations which are $<_\mathcal{O}n$. But different notations for the same ordinal may be incomparable in $\mathcal{O}$ (essentially: we can't tell in general whether two "computable descriptions" describe the same ordinal), and the associated functions can be wildly different.
It's not necessarily worth digging too deeply into the precise definition of $\mathcal{O}$; it's enough to think of it as a set of indices for computable ordinals which is "reasonably nice", and that will give you the right intuition. The specific definition is extremely useful for proving theorems, but those theorems only show up a ways down the road. For now, for example, we could just work for indices of Turing machines which happen to describe ordinals.
Now, usually we get around this by developing the fast-growing hierarchy below some specific "natural" notation for some fixed ordinal - e.g., the Lob-Wainer hierarchy works below a reasonable notation for $\epsilon_0$ - and in this case, Stella Biderman's answer is correct, that for any fixed ordinal notation we can of course diagonalize away from it. However, if you want to consider the fully general fast-growing hierarchy defined on all of $\mathcal{O}$, then it turns out that there is no escape! In fact, for any computable function $g$, I can come up with an ordinal notation $n_g$ for $\omega^2$ such that $f_{n_g}$ grows faster than $g$ (and such an $n_g$ can be computably gotten from (an index for) $g$!).
(This phenomenon - that $\omega^2$ is already enough - happens with annoying frequency when playing around with $\mathcal{O}$. Once we get to large enough (really, very small!) ordinals, there are "sufficiently weird" notations that do whatever you want. For instance, the Ershov hierarchy (link to be added once I find it :P) for sets computable from the Halting Problem has to be defined in terms of notations, since in terms of ordinals it collapses quite quickly.)
