We investigate the existence and nature of classical maximum force bound between two black holes with touching horizons. Besides general relativity, the maximum force bound is independent of black hole masses only in Moffat's theory, Brans Dicke theory (which is the same as Einstein's for vacuum) and the higher dimensional generalization of Einstein's theory, pure Lovelock gravity which is characterised by having single $n$th order term in Lovelock polynomial without sum over lower orders in the action. Further if the bound is to exist in higher dimensions and is entirely in terms of the velocity of light and the gravitational constant, it has uniquely to be pure Lovelock gravity. In pure Lovelock gravity, the maximum force bound exists in all $3n$-space dimensions and has the value, $c^4/4G_n$ where $G_n$ is the corresponding gravitational constant. The absence of mass dependence in the maximum force bound may have relevance for the formation of naked singularities.

10 pages, no figures, section on power law gravity dropped, additions to conclusions, The paper is dedicated to the fond memory of one of the authors, John Barrow who passed away on 26 Sept 2020 - Naresh Dadhich, coauthor