Iterated Taylor series

[Xakoro]

Suppose you have a taylor series (ie sinx= x-(x^3/3!)+(x^5/5!)-(x^7/7!)+... to infinity).
You then iterate it.
ie iterated once gives (sin(sinx) = sinx-(sin^3x/3!)+(sin^5x/5!)-... to infinity.)
How would you find the limit of it iterated infinitely many times?
(Yes I know that for sinx you get a square wave)
As an example- log(sin(log(sin...?

[DarkBeam]

... an hybrid fractal ...