  Given a closed simply connected manifold $M$ of dimension $2n\ge6$, 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 $M\sharp\Sigma$ 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.
