  Standard-Model Higgs bosons are dominantly produced via the gluon-fusion
mechanism $gg \to H$ at the LHC, i.e. in a loop-mediated process with top loops
providing the dominant contribution. For the measured Higgs boson mass of $\sim
125$ GeV the limit of heavy top quarks provides a reliable approximation as
long as the relative QCD corrections are scaled with the full mass-dependent LO
cross section. In this limit the Higgs coupling to gluons can be described by
an effective Lagrangian. The same approach can also be applied to the coupling
of more than one Higgs boson to gluons. We will derive the effective Lagrangian
for multi-Higgs couplings to gluons up to N$^4$LO thus extending previous
results for more than one Higgs boson. Moreover we discuss gluonic Higgs
couplings up to NNLO, if several heavy quarks contribute.
