A search for beyond-the-standard-model neutral Higgs bosons decaying to a pair of bottom quarks, and produced in association with at least one additional bottom quark, is performed with the CMS detector. The data were recorded in proton-proton collisions at a centre-of-mass energy of 13 TeV at the CERN LHC and correspond to an integrated luminosity of 36.7–126.9 fb−1, depending on the probed mass range. No signal above the standard model background expectation is observed. Upper limits on the production cross section times branching fraction are set for Higgs bosons in the mass range of 125–1800 GeV. The results are interpreted in benchmark scenarios of the minimal supersymmetric standard model, as well as suitable classes of two-Higgs-doublet models.