We introduce the notion of Bohr chaoticity, which is a topological invariant, and is opposite to the property required by Sarnak's conjecture. Such a system is by definition never orthogonal to any non-trivial weight and it must be of positive entropy. But having positive entropy is not sufficient to ensure the Bohr chaoticity. We prove the Bohr chaoticity for all toral affine dynamical systems of positive entropy, all subshifts of finite type of positive entropy and all \beta-shifts. However, uniquely ergodic dynamical systems are not Bohr chaotic and there are many such dynamical systems of positive entropy. This is a joint work with Aihua FAN and Weixiao SHEN.