  The volume of on-shell D=4, N=8 superspace is shown to vanish. Despite this,
it is shown that there is a fully supersymmetric and duality-invariant
candidate d^8 R^4 counterterm corresponding to an anticipated seven-loop
logarithmic divergence in D=4. We construct this counterterm explicitly and
also give the complete nonlinear extension of the 1/8-BPS d^6 R^4 invariant.
Similar results are derived for N=4,5 & 6.
