GCH. The standard forcing of the GCH is an Easton-support iteration that simply iteratively forces the next instance directly, and this forcing has very nice structural features that will enable to preserve most all of the usual large cardinals by the usual lifting arguments. The usual master condition arguments, the same arguments used in the Laver preparation, show that this forcing will preserve all supercompact cardinals, and one can use the same methods to preserve larger large cardinals. For example, every measurable cardinal, every strong cardinal, every supercompact cardinal, almost every partiall supercompact cardinal, every rank-to-rank cardinal, and so forth, is preserved by the canonical forcing of the GCH. So the GCH is relatively consistent with most all of the usually considered large cardinals.
Ground Axiom. The ground axiom is similarly consistent with large cardinals. This follows by the main dissertation result Jonas Reitz, who was my student at the time.
Hamkins, Joel David; Reitz, Jonas; Woodin, W. Hugh, The ground axiom is consistent with (V \neq \text{HOD}), Proc. Am. Math. Soc. 136, No. 8, 2943-2949 (2008). ZBL1145.03029.