The rate for Higgs ( H ) bosons production in association with either one ( t H ) or two ( t t ¯ H ) top quarks is measured in final states containing multiple electrons, muons, or tau leptons decaying to hadrons and a neutrino, using proton-proton collisions recorded at a center-of-mass energy of 13 TeV by the CMS experiment. The analyzed data correspond to an integrated luminosity of 137 fb - 1 . The analysis is aimed at events that contain H → W W , H → τ τ , or H → Z Z decays and each of the top quark(s) decays either to lepton+jets or all-jet channels. Sensitivity to signal is maximized by including ten signatures in the analysis, depending on the lepton multiplicity. The separation among t H , t t ¯ H , and the backgrounds is enhanced through machine-learning techniques and matrix-element methods. The measured production rates for the t t ¯ H and t H signals correspond to 0.92 ± 0.19 (stat) - 0.13 + 0.17 (syst) and 5.7 ± 2.7 (stat) ± 3.0 (syst) of their respective standard model (SM) expectations. The corresponding observed (expected) significance amounts to 4.7 (5.2) standard deviations for t t ¯ H , and to 1.4 (0.3) for t H production. Assuming that the Higgs boson coupling to the tau lepton is equal in strength to its expectation in the SM, the coupling y t of the Higgs boson to the top quark divided by its SM expectation, κ t = y t / y t SM , is constrained to be within - 0.9 < κ t < - 0.7 or 0.7 < κ t < 1.1 , at 95% confidence level. This result is the most sensitive measurement of the t t ¯ H production rate to date.