the class $\mathfrak{F}_k$ of the fast growing hierarchy is the closure under substitution and limited recursion of the constant, sum,projections and $F_n$ functions for $n\leq k,$where $F_n$ is defined recursively by $ \begin{eqnarray*} F_0(x) &\triangleq & x+1\\ F_{n+1}(x) &\triangleq &F_n^{x+1}(x) \end{eqnarray*}$
Here, $F_n^{x+1}(x)=\underbrace{F_n(F_n(\cdots (F_n}_{x+1}(x)))$
The hierarchy is strict for $k\geq 1$, i.e. $\mathfrak{F}_{k}\subsetneq \mathfrak{F}_{k+1}$.
Then, if a function $g$ can be written as $ \begin{eqnarray} g=\underbrace{f_1^{f_2^{.^{.^{.{f_L}}}}}}_{L} \end{eqnarray}$
Here $f_i,L$ are functions with variable $x$, and all belong to$\mathfrak{F}_3,$ then which class does $g$ belong to? Is $g\in \mathfrak{F}_4$?
Note: you can find more information on "the fast growing hierarchy" on page 9 in this following paper: http://arxiv.org/abs/1007.2989