For every Hausdorff function h, the paper gives a construction of a compact metric space of infinite diameter-based h-packing measure which has no subsets of positive finite measure. It then indicates how such a construction can be modified to provide the same result for a radius-based packing measure in the case of certain Hausdorff functions which do not satisfy a doubling condition. This is in surprising contrast to another result which shows that for a particular radius-based definition of packing measure no such example can be found.