  We study closure properties of measurable ultrapowers with respect to
Hamkin's notion of "freshness" and show that the extent of these properties
highly depends on the combinatorial properties of the underlying model of set
theory. In one direction, a result of Sakai shows that, by collapsing a
strongly compact cardinal to become the double successor of a measurable
cardinal, it is possible to obtain a model of set theory in which such
ultrapowers possess the strongest possible closure properties. In the other
direction, we use various square principles to show that measurable ultrapowers
of canonical inner models only possess the minimal amount of closure
properties. In addition, the techniques developed in the proofs of these
results also allow us to derive statements about the consistency strength of
the existence of measurable ultrapowers with non-minimal closure properties.
