Are large cardinals about more than just consistency? The other day, I was reading the preface of Kanamori's The Higher Infinite and noticed that he says large cardinals provide a useful 'measuring stick' for consistency. That raised the question of whether large cardinals are motivated simply by searching for their consistency strengths or if there are other motivations. If large cardinals have appeal outside of consistency strength, like mathematical beauty or interesting properties, are there any examples that someone with a moderate knowledge of set theory could understand?
Related question on MSE
 A: Asaf Karagila already wrote an excellent answer at math.SE, but here is a simple point that may be helpful.  In any area of math, the natural course of research leads one to ask questions, pose conjectures (and prove them if possible), and impose extra structure on the objects of study if that seems to be necessary for them to have desirable properties.  Set theory is not any different in this regard, except that in set theory, we can often detect that some of our "conjectures" are hopeless, in the sense that Gödel's incompleteness theorems tell us that we cannot possibly prove them in the conventional mathematical sense.  So you can think of large cardinal axioms as "natural conjectures," except that we don't call them conjectures because we know we can't prove them.  Alternatively, you can think of increasingly strong large cardinal axioms as imposing increasing amounts of structure on $V$, giving it extra desirable properties.
One can still ask, if we're just looking for "natural conjectures," why do large cardinal axioms stand out?  Why aren't there lots of other, equally prominent axioms?  Well, as a matter of fact, there are other axioms that set theorists study, but large cardinals stand out because they somewhat miraculously seem to line up nicely in a single hierarchy.  This fact is not at all obvious a priori (for example, it is not at all obvious from the definition of a measurable cardinal that it must be inaccessible).  To some people, it suggests that the large cardinal axioms are "true," but even if you don't believe in truth in this sense, the large cardinal axioms bring a lot of order to what might otherwise be a chaotic universe.  Instead of a huge mess of "natural conjectures," we find that many other questions can be more-or-less answered by showing that they lie sandwiched in between two well-known large cardinal axioms.
