Let $P \in \mathbb{Q}[X]$, is it true that $P \circ P \in \mathbb{Z}[X]$ and $P \circ P \circ P \in \mathbb{Z}[X]$ implies $P \in \mathbb{Z}[X]$ ?
The two conditions are necessary because, if $P=2X^2-\frac{1}{2}$, we have $P \circ P \in \mathbb{Z}[X]$ and $P \circ P \circ P \notin \mathbb{Z}[X]$, but $P \notin \mathbb{Z}[X]$. And if $P=9X^3+\frac{1}{3}$, we have $P \circ P \circ P \in \mathbb{Z}[X]$ and $P \circ P \notin \mathbb{Z}[X]$, but $P \notin \mathbb{Z}[X]$.
