Consider the set of all functions of one variable $x\in[0,1]$ that can be constructed from any number of instances of that variable using parentheses and exponentiation only:
$$x,\;x^x,\,x^{x^x},\;\left(x^x\right)^x,\;x^{x^{x^x}},\;x^{\left(x^x\right)^x},\;\left(x^x\right)^{x^x},\;\left(\left(x^x\right)^x\right)^x,\; x^{x^{x^{x^x}}},\;x^{x^{\left(x^x\right)^x}},\;...$$
Here are graphs of some functions from this set:
Looking at these graphs made me think — what is the supremum of arc lengths of these graphs on $[0,1]$? Is there a closed form expression for it? Is there an efficient algorithm that can compute it to an arbitrary precision?