How can one tell if a group element is a k-fold nested commutator? A computable invariant of words in groups that does not vanish on k-fold commutators will help. For free groups this is achieved by Fox calculus, whose geometric applications include Milnor invariants of links, and there are generalizations for braid groups and RAAGs, but beyond that little is known. We introduce a complete and computable collection of such invariants for any group, using the algebraic Bar construction. Consequences of this theory include nonvanishing of mod p Massey products for finite groups, and a dual version of the Johnson homomorphism defined for automorphisms of arbitrary groups.