An integer packing set is a set of non-negative integer vectors with the property that, if a vector x is in the set, then every non-negative integer vector y with y≤x is in the set as well. The main result of this paper is that integer packing sets, ordered by inclusion, form a well-quasi-ordering. This result allows us to answer a recently posed question: the k-aggregation closure of any packing polyhedron is again a packing polyhedron.