Continuous iteration of exponentation

The continuous iteration of the exponentiation operation may hold the key to extending transseries to deal with larger and larger growth rates. We can iterate the exponentation operation to integer arguments for example $exp^2(x) = exp(exp(x))$, $exp^{-1}(x) = log(x)$, and $exp^{-2}(x) = log(log(x))$ so the key to continuous iteration are the fractional iterates of exponentiation such as the half-iterate.

These fractional iterates of exponentiation apparently can be described by the natural tetration operation $tet$ and its inverse function in a similar manner to how we can describe general iterates of multiplication using $exp$ and its inverse function $ln$. As a result of this it may be the case that we should extend transseries with the $tet$ function in order to implement iterates of $exp$ and larger growth rates.