The CMS detector at the LHC has recorded events from proton-proton collisions, with muon momenta reaching up to 1.8 TeV in the collected dimuon samples. These high-momentum muons allow direct access to new regimes in physics beyond the standard model. Because the physics and reconstruction of these muons are different from those of their lower-momentum counterparts, this paper presents for the first time dedicated studies of efficiencies, momentum assignment, resolution, scale, and showering of very high momentum muons produced at the LHC. These studies are performed using the 2016 and 2017 data sets of proton-proton collisions at $\sqrt{s} =$ 13 TeV with integrated luminosities of 36.3 and 42.1 fb$^{-1}$, respectively.