I've heard before that Knuth holds the record for the largest constant used in a mathematical proof.
I was wondering what is the largest cardinal ever explicitly considered in set theory. I presume this will depend on the axioms chosen, but I'd be interested to know what's the 'largest infinity conceived by man'.
The question is a bit vague, for some reasons.
For example, in many consistency proofs in set theory one begins by taking some very very very large cardinal $\kappa$. This is not only an assumption which tells us that the cardinality of $\kappa$ is ineffable and beyond our wildest imagination; it also means that assuming that such cardinal exists is a far stronger assumption than just the axioms of $\sf ZFC$.
One recent work I've been doing with a classmate/colleague (we have the same advisor) begins with a Mahlo cardinal above a supercompact cardinal. While Mahlo are not "that large" of a large cardinal, supercompacts are, and so this Mahlo cardinal is quite big. But you can just as well begin with a Mahlo cardinal which is larger than an infinite number of supercompact cardinals, which is an even larger cardinal. And you can keep pushing this sort of limit, on and on and on. Of course, one of the aims is to be minimal in the assumptions, but this sort of "pushing up" is naturally occurring in set theory (as is "pushing down" the consistency bounds and attempting to find minimal assumptions).
So in a kinda of copping out, I'll say that the largest infinity conceived is the universe of set theory. That is an infinity so large we cannot even assign it a cardinal number. It is larger than all the other infinities, by definition. And within the bounds of a fixed universe, one cannot possibly exceed it in any way.
Of course, this depends on what universe of set theory you're in, and whether or not you prescribe to a multiverse approach, or you're a Platonist, or so on.