In the mathematical theory of the infinite, classes of cardinals which are very remote have been considered, with names such as inaccessible cardinals, strongly inaccessible cardinals, Mahlo cardinals, hyper-Mahlo cardinals, and so on.
While I will not begin to attempt to touch on much of this here, I have encountered a way to provide a small taste of some of this.
A few simple axioms suffice to define a set the number of whose elements must be a strongly inaccessible cardinal, unless it has either no elements or aleph-null elements. This kind of set is defined as a Grothendieck universe.
The axioms which define such a universe are:
Note that the fourth axiom does not state that set B is an element of the universe, although that is derivable from the four axioms as well.
If we assume the truth of the Generalized Continuum Hypothesis, we might be able to begin to approach assigning a definite value to the minimum number of elements a Grothendieck universe containing the set of real numbers might have. Or perhaps we should start with only the integers? But the power set of the integers has the same cardinality as the real numbers, and so it doesn't seem to matter which of those starting points we choose.
Assuming the GCH, and given that power sets of power sets belong to such a universe, we are already beyond aleph-n for any finite n.
So, one might timidly ask if perhaps its cardinality might be aleph-omega. But it appears that the fourth axiom will not allow that, as it vastly increases the size of the universe; and, indeed, aleph-omega is too small to be a strongly inaccessible cardinal.