  The spectral sequence associated to the Arone-Goodwillie tower for the n-fold
loop space functor is used to show that the first two non-trivial layers of the
nilpotent filtration of the reduced mod 2 cohomology of a (sufficiently
connected) space with nilpotent cohomology are comparable. This relies upon the
theory of unstable modules over the mod 2 Steenrod algebra, together with
properties of a generalized class of almost unstable modules which is
introduced here.
  An essential ingredient of the proof is a non-vanishing result for certain
extension groups in the category of unstable modules localized away from
nilpotents.
