A non-principal ultrafilter $\mathcal{U}$ on $\omega$ is called selective (or Ramsey) if for every partition $\mathcal{P}$ of $\omega$ disjoint with $\mathcal{U}$ there is $A\in\mathcal{U}$ such that $|A\cap B|\le1$ for every $B\in\mathcal{P}$. There are several weakenings of this definition, but still expressing some kind of selectivity; most notably P-points and Q-points.
My question is as follows. Do you know any notion of highly non-selective ultrafilters? A class of ultrafilters about which we can say that they are very far from being selective? Or maybe some way of measuring how ultrafilters are selective?
Selective ultrafilters are exactly the minimal points of the Rudin-Keisler ordering of non-principal ultrafilters on $\omega$, so any ultrafilter which is not a minimal one is not selective. But how can we see this ordering in the context of selectivity?
I know that the question is vague, but maybe you have some association with it. I will appreciate any answer.