Is there a mathematical term, practical application, or area of math that covers a function raised to itself? 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.

$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.
