Destabilizing amalgamated Heegaard splittings
  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. 
Jennifer Schultens, Richard Weidmann