  The Milnor degree of a 3-manifold is an invariant that records the maximum
simplicity, in terms of higher order linking, of any link in the 3-sphere that
can be surgered to give the manifold. This invariant is investigated in the
context of torsion linking forms, nilpotent quotients of the fundamental group,
Massey products and quantum invariants, and the existence of 3-manifolds with
any prescribed Milnor degree and first Betti number is established.
  Along the way, it is shown that the number M(k,r) of linearly independent
Milnor invariants of degree k, distinguishing r-component links in the 3-sphere
whose lower degree invariants vanish, is positive except in the classically
known cases (when r = 1, and when r = 2 with k = 2, 4 or 6).
