We consider integer sequences that satisfy a recursion of the form x_n+1 = P(x_n) for some polynomial P of degree d > 1. If such a sequence tends to infinity, then it satisfies an asymptotic formula of the form x_n ∼ A α^dn, but little can be said about the constant α. In this paper, we show that α is always irrational or an integer. In fact, we prove a stronger statement: if a sequence (G_n)_{n ≥ 0} satisfies an asymptotic formula of the form G_n = A αⁿ + B + O(α^{-ε n}), where A, B are algebraic and α > 1, and the sequence contains infinitely many integers, then α is irrational or an integer.