  In 1952, Littlewood stated a conjecture about the average growth of spherical
derivatives of polynomials, and showed that it would imply that for entire
function of finite order, "most" preimages of almost all points are
concentrated in a small subset of the plane. In 1988, Lewis and Wu proved
Littlewood's conjecture. Using techniques from complex dynamics, we construct
entire functions of finite order with a bounded set of singular values for
which the set of exceptional preimages is infinite, with logarithmically
growing cardinality.
