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Considering the iteration of exponentiation: $x$, $b^x$, $b^{(b^x)}$ $ \cdots$ we might call the number of repeated appearances of the base-parameter $b$ the height $h$ of the exponential-/powertower. Usually the top-exponent parameter $x$ is taken as $1$ (or $b$ which then shifts simply the height index by 1) and this is then written as b^^h.

If the base-parameter b is a natural number we discuss questions like "What is the last digit of $2^{2^{2^{2^2}}}$ ?" or similar modular questions. For positive real bases there are common problems like "For what real base b does the infinite exponentialtower converge?" (which was already answered by Euler and Goldbach and others) For complex bases this gets then many more interesting properties and a rich structure similarly to, for instance, the famous Mandelbrot fractals.

"Tetration" in a more precise sense, is then, if the (iteration) height parameter is considered the interesting parameter, and especially if understood as rational or even continuous (real or complex) variable. Prominent questions are "what is the half-iterate of $\exp(x)$?" or differently formulated "What is the function $f(x)$, such that $f(f(x))=\exp(x)$ , does it have a power series and does this have a nonzero radius of convergence?" . Questions of analycity, radius of convergence, uniqueness of the found/proposed solutions are some of the core difficulties for which answers are sought.

In generalization of the idea of iteration to functions other than the exponential-function, say the sine/cosine-functions, squarerroot or anything else, it is common to call the function, which describes the iteration depending on the iteration height, as "superfunction", so you shall find questions related to "tetration" if you search for the keyword "superfunction" as well.

This idea of iteration can be applied to also to tetration/iterated-exponentiation itself, this leads naturally to the concept of the .

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