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On the existence/applications of infinitely-nested functions

Inside a previous question, one particular nested function shown is the known tetration. This "kind" of arbitrary repeated functions has always intrigued me, because inside their properties lie so ...
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Evaluating iterated sine function

Let $f(x,1)=\sin(x)$ and $f(x,i)=f(\sin(x),i-1)$ ($f$ is the iterated sine function). For arbitrary $N$,$x_0$, how quickly can $f(x_0,N)$ be computed? Answer to this question discusses ...