Abstract: Following work of Bugeaud, Corvaja, and Zannier for integers, Ailon and Rudnick prove that for any multiplicatively independent polynomials, $a, b \in {\mathbb C}[x]$, there is a polynomial $h$ such that for all $n$, we have $\gcd(a^n - 1, b^n - 1) \mid h.$ We prove a compositional analog of this theorem, namely that if $f, g \in {\mathbb C}[x]$ are nonconstant compositionally independent polynomials and $c(x) \in {\mathbb C}[x]$, then there are at most finitely many $\lambda\in {\mathbb C}$ with the property that there is an $n$ such that $(x-\lambda)$ divides $\gcd(f^{\circ n}(x) - c(x), g^{\circ n}(x) - c(x))$. This is joint work with Tom Tucker.