I will present for each integer d > 3 a K-trivial log canonical variety over the complex numbers of dimension d that does not admit a Beauville-Bogomolov decomposition. That is, for the universal cover X of the variety, there is no decomposition of X as a product of an affine space and of three types of projective varieties: strict Calabi-Yau, symplectic and rationally connected varieties. Note: the counterexample is sharp in the sense that for Kawamata log terminal varieties the decomposition does hold.[-]