# Supremum of arc lengths of graphs of power towers

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?

• A wild guess would be $1+\sqrt 2$. – Szeto Mar 5 at 22:41