Lattices with many Borcherds products. We prove that there are only finitely many isometry classes of even lattices ( L) of signature ( (2,n)) for which the space of cusp forms of weight ( 1+n/2) for the Weil representation of the discriminant group of ( L) is trivial. We compute the list of these lattices. They have the property that every Heegner divisor for the orthogonal group of ( L) can be realized as the divisor of a Borcherds product. We obtain similar classification results in greater generality for finite quadratic modules.