What techniques exist for working with "iterated" functions such as f(f(x))?

Question

I keep encountering questions that sound like obscure puzzles, involving expressions such as $f(f(x))$. I don't know of any techniques for working with such things, or getting from knowledge of $f(f(x))$ to knowledge of $f(x)$, other than guesswork. Is there a subject that covers this topic? A reference text? Or is it simply a random handful of off-the-wall questions with no general theory?