This Letter describes a search for Higgs boson pair production using the combined results from four final states: $bb\ensuremath{\gamma}\ensuremath{\gamma}$, $bb\ensuremath{\tau}\ensuremath{\tau}$, $bbbb$, and $bbVV$, where $V$ represents a $W$ or $Z$ boson. The search is performed using data collected in 2016 by the CMS experiment from LHC proton-proton collisions at $\sqrt{s}=13\text{ }\text{ }\mathrm{TeV}$, corresponding to an integrated luminosity of $35.9\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$. Limits are set on the Higgs boson pair production cross section. A 95% confidence level observed (expected) upper limit on the nonresonant production cross section is set at 22.2 (12.8) times the standard model value. A search for narrow resonances decaying to Higgs boson pairs is also performed in the mass range 250--3000 GeV. No evidence for a signal is observed, and upper limits are set on the resonance production cross section.