The Laplacian Δ g of a compact Riemannian manifold ( M , g ) is called maximally degenerate if its eigenvalue multiplicity function m g ( k ) is of maximal growth among metrics of the same dimension and volume. Canonical spheres ( S n , can ) and CROSSes are MD, and one asks if they are the only examples. We show that a MD metric must be at least a Zoll metric with just one distinct eigenvalue in each cluster, and hence with all band invariants equal to zero. The principal band invariant is then calculated in terms of geodesic integrals of curvature and Jacobi fields, giving a local metric condition for maximal degeneracy. In special cases (surfaces of revolution, real projective spaces) the MD metrics are shown to be CROSSes.