  We construct a family of transcendental entire functions whose Julia sets
have packing dimension in $(1,2)$. These are the first examples where the
computed packing dimension is not $1$ or $2$. Our construction will allow us
further show that the set of packing dimensions attained is dense in the
interval $(1,2)$, and that the Hausdorff dimension of the Julia sets can be
made arbitrarily close to the corresponding packing dimension.
