In this work we study the univariate and multivariate quantitative approximation by multi-composite Kantorovich–Choquet-type quasi-interpolation neural network operators with respect to the supremum norm. This is achieved with rates via the first univariate and multivariate moduli of continuity. We approximate continuous and bounded non-negative functions on RN,N∈N. When they are also uniformly continuous we have pointwise and uniform convergences, plus Lp estimates. Our multi-composite activation functions are formed by general sigmoid functions.