Is there a closed form$^*$ for the following recursive function: \begin{align} f_0(x) &:= x\\ f_{n+1}(x) &:= f_n(x)^{f_n(x)} \end{align} Assuming that $x$ is an element of a semi-ring$^\dagger$, with exponentiation defined canonically.
The first few $f_i$ functions:\begin{align}
f_0(x) &= x\\
f_1(x) &= x^x\\
f_2(x) &= x^{x^{x+1}}\\
f_3(x) &= x^{x^{x^{x+1}+x+1}}\\
\end{align}
$*$: That is in terms of elementary functions, or as a sequence $(a,b,c,...)$ denoting $a^{b^{c^...}}$, or both.
$\dagger$: If $x$ needs to be more constrained; eg, element of a ring, a field, etc...; note it in your answer, please.