The class of f-rings in which the product of any n elements is comparable to zero (n-orderpotent f-rings) generalizes the concept of both totally ordered and nilpotent f-rings. Necessary and sufficient conditions are found for an f-ring to be n-orderpotent. It is shown that n-orderpotency is closely related to the ring having sufficiently many annihilating elements. Special consideration is given to generalized semigroup rings, a rich source of examples.