  We construct a sequence of pairs of 3-manifolds each with torus boundary and
with the following two properties:
  1) For the result of a carefully chosen glueing of the nth pair of
3-manifolds along their boundary tori, the ratio of the genus of the resulting
3-manifold to the sum of the genera of the pair of 3-manifolds is less than
1/2.
  2) The result of amalgamating certain unstabilized Heegaard splittings of the
pair of 3-manifolds to form a Heegaard splitting of the resulting 3-manifold
produces a stabilized Heegaard splitting that can be destabilized successively
n times.
