High-multiplicity pp collisions exhibit features, traditionally associated with nuclear effects. Coherence motivates to treat high-multiplicity pp, pA, and AA collisions on an equal footing. We rely on the phenomenological parametrization for mean multiplicities of light hadrons and J/ψ, assuming their linear dependence on Ncoll in pA collisions. The results of this approach underestimate the recently measured production rate of J/ψ at very high hadronic multiplicities. The linear dependence of J/ψ multiplicity on Ncoll is subject to predicted nonlinear corrections, related to mutual boosting of the saturation scales in colliding dense parton clouds. A parameter-free calculation of the nonlinear corrections allows us to explain data for pT-integrated yield of J/ψ at high hadronic multiplicities. Calculations are in a good accord with data binned in several pTintervals as well. As was predicted, Υ and J/ψ are equally suppressed at forward rapidities in pA collisions. Consequently, their fractional multiplicities at forward rapidities in pp collisions are equal as well, and their magnitude agrees with data.