A search for new physics in energetic, high-multiplicity final states has been performed using proton-proton collision data collected with the CMS detector at a center-of-mass energy of 13 TeV and corresponding to an integrated luminosity of 2.3 inverse femtobarns. The standard model background, dominated by multijet production, is determined exclusively from control regions in data. No statistically significant excess of events is observed. Model-independent limits on the product of the cross section and the acceptance of a new physics signal in these final states are set and further interpreted in terms of limits on the production of black holes. Semiclassical black holes and string balls with masses as high as 9.5 TeV, and quantum black holes with masses as high as 9.0 TeV are excluded by this search in the context of models with extra dimensions, thus significantly extending limits set at a center-of-mass energy of 8 TeV with the LHC Run 1 data.