Are there any applications of the largest large cardinals to consistency results concerning, say, cardinals below $\aleph_{\aleph_\omega}$? Or perhaps to prove results in descriptive set theory? I am thinking of ZFC + Axiom I0, ZF + Reinhardt, ZF + Berkeley. See here for definitions. The largest cardinal I know of to be used for a consistency result is a 2-huge cardinal to prove the consistency of $(\aleph_3,\aleph_2,\aleph_1) \twoheadrightarrow (\aleph_2,\aleph_1,\aleph_0)$.

I would be remiss not to mention the results of Laver about left-distributive algebras proved using Axiom I3. Some of the results have not been brought down to ZFC. This is remarkable and I suppose answers the question as posed, but my motivation is to see if there are forcing constructions aimed at small cardinals that use these very large cardinals.