This question is inspired by a [riddle][1] in math.stackexchange.

Let $P$ be a polynomial, and $O = \{P^{(n)}(0) : n \geq 0\}$ be its orbit under zero (viewed as a set). Suppose that $O$ contains infinitely many integers. Is it true that for some $n$, $P^{(n)}$ is a polynomial with integral coefficients?

We can ask the same question replacing integers with rationals.

EDIT: Nick and David gave simple counterexamples for the first question.
Still open:

 1. In the setting of the original question, is it true that some composition power of $P$ takes integers to integers?
 2. The original question with rationals.

  [1]: https://math.stackexchange.com/questions/8101/iterated-polynomial-problem