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