User blog comment:Vel!/Ultracardinals/@comment-5529393-20140619121839

The weak ultracardinals are precisely the regular cardinals. If k is a regular cardinal, then cof(k) = k, and if a, b < k, then a + b, ab, a^b, and cof(a) are less than k.

The ultracardinals are defined to be cardinals such that no weak ultracardinal is bigger. Obviously, there are no such cardinals.