Infinite compositions of functions

Summary

An infinite composition is the limit of the finite compositions $f_1 \circ f_2 \circ \cdots \circ f_n$, or of $f_n \circ \cdots \circ f_1$, of a sequence of functions. It extends function composition to infinitely many factors, in the same way as a series extends a finite sum. Continued fractions, nested radicals and power towers are all infinite compositions. Iteration of a single function is the special case where all $f_n$ are equal.