  There are several examples in the literature showing that compactness-like
properties of a cardinal $\kappa$ cause poor behavior of some generic
ultrapowers which have critical point $\kappa$ (Burke \cite{MR1472122} when
$\kappa$ is a supercompact cardinal; Foreman-Magidor \cite{MR1359154} when
$\kappa = \omega_2$ in the presence of strong forcing axioms). We prove more
instances of this phenomenon. First, the Reflection Principle (RP) implies that
if $\vec{\mathcal{I}}$ is a tower of ideals which concentrates on the class
$GIC_{\omega_1}$ of $\omega_1$-guessing, internally club sets, then
$\vec{\mathcal{I}}$ is not presaturated (a set is $\omega_1$-guessing iff its
transitive collapse has the $\omega_1$-approximation property as defined in
Hamkins \cite{MR2540935}). This theorem, combined with work from \cite{VW_ISP},
shows that if $PFA^+$ or $MM$ holds and there is an inaccessible cardinal, then
there is a tower with critical point $\omega_2$ which is not presaturated;
moreover this tower is significantly different from the non-presaturated tower
already known (by Foreman-Magidor \cite{MR1359154}) to exist in all models of
Martin's Maximum. The conjunction of the Strong Reflection Principle (SRP) and
the Tree Property at $\omega_2$ has similar implications for towers of ideals
which concentrate on the wider class $GIS_{\omega_1}$ of $\omega_1$-guessing,
internally stationary sets.
  Finally, we show that the word "presaturated" cannot be replaced by
"precipitous" in the theorems above: Martin's Maximum (which implies SRP and
the Tree Property at $\omega_2$) is consistent with a precipitous tower on
$GIC_{\omega_1}$.
