Some abstract examples would be: $f(x)^{f(x)}$ or $f(x)^{f(x)^{f(x)...}}$ Actual equations I've attempted to look at can be viewed here on desmos.com There seems to be a pattern of common convergence, and a relation to the functions raised to a constant power, but in my searches I have yet to find any meaning or usage.

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    Wikipedia calls it tetration–and it lists other coined terms. $f(x)^{f(x)^{f(x)\cdots}}$ can be written compactly as $^nf(x)$. – Antonio DJC Feb 23 '17 at 4:55

$f(x) = \sin(x) \quad \rightarrow \quad f(x)^{f(x)} $

Produces a linear repetition of an angry man with a sweet moustache ...

That's literally the only usage I've ever come up with.

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