In what I write below, by "ultrafilter" I mean a non-principal ultrafilter.
Given an ultrafilter $U$ on some set $S$, let $\mu$ be the least cardinal such that $U$ is $\mu$-complete but not $\mu^+$-complete. Call this number the completeness number of $U$. It is easy to check that this must always be an infinite regular cardinal.
The question is which cardinals can appear as completeness numbers for ultrafilters. Stated otherwise: for which cardinals $\mu$ we can find examples of ultrafilters that have completeness number $\mu$.
For instance, if an ultrafilter is not countably complete, then its completeness number is $\aleph_0$ (and if there are no measurable cardinals then in fact every ultrafilter has completeness number $\aleph_0$). This means that $\aleph_0$ can be completeness number.
What about $\aleph_1$? or $\aleph_n$? or any other regular or even large cardinal $\mu$?
Perhaps this is well-known but I cannot find a reference.