Suppose f ∈ K[x] is a polynomial. The absolute Galois group of K acts on the preimage tree T of 0 under f. The resulting homomorphism φf : GalK → Aut T is called the arboreal Galois representation. Odoni conjectured that for all Hilbertian fields K there exists a polynomial f for which φf is surjective. We show that this conjecture is false.