Given a closed simply connected manifold M of dimension 2nge 6, we compare the ring of characteristic classes of smooth oriented bundles with fibre M to the analogous ring resulting from replacing M by the connected sum Msharp Sigma with an exotic sphere Sigma . We show that, after inverting the order of Sigma in the group of homotopy spheres, the two rings in question are isomorphic in a range of degrees. Furthermore, we construct infinite families of examples witnessing that inverting the order of Sigma is necessary.