I was thinking about infinite exponential representation of real numbers (like $2=e^{e^{-e^{-e^{e^{-e^{e^{e^{-e^{-e^{-e^{-e^{-e^{e^{-e^{e^{e^{-e^{e^{\cdot^{\cdot^{\cdot}}}}}}}}}}}}}}}}}}}}}$. The sequence of signs before exponents can be obtained by repeated application of $\ln|x|$ to $2$ and taking a sign of each result. It seems that this gives an almost 1-1 correspondence between $\mathbb{R}$ and the set of infinite sequences of signs (or positive and negative 1's) (except that $0$ has two representations $\pm e^{-e^{e^{e^{\cdot^{\cdot^{\cdot}}}}}}$ and sequences $\pm e^{e^{e^{\cdot^{\cdot^{\cdot}}}}}$ diverge to $\pm \infty$).

Has this representation been studied? Does any algebraic number (except $0$, $1$ and $-1$) have an eventually periodic representation? What we can say about frequency of each sign in representation of a particular number (say, $2$)? (first several hundreds of elements suggest that $-1$ appears two times more often than $1$). Are there arbitrarily long runs of the same sign?