A search is presented for the production of a pair of Higgs bosons, where one decays into two photons and the other one into a bottom quark-antiquark pair. The analysis is performed using proton-proton collision data at $\sqrt{s} =$ 13 TeV recorded in 2016 by the CMS detector at the LHC, corresponding to an integrated luminosity of 35.9 fb$^{-1}$. The results are in agreement with standard model (SM) predictions. In a search for resonant production, upper limits are set on the cross section for new spin-0 or spin-2 particles. For the SM-like nonresonant production hypothesis, the data exclude a product of cross section and branching fraction larger than 2.0 fb at 95% confidence level (CL), corresponding to about 24 times the SM prediction. Values of the effective Higgs boson self-coupling $\kappa_\lambda$ are constrained to be within the range $-$11 < $\kappa_\lambda$ < 17 at 95% CL, assuming all other Higgs boson couplings are at their SM value. The constraints on $\kappa_\lambda$ are the most restrictive to date.