  We construct families of pairs of Heegaard splittings that must be stabilized
several times to become equivalent. The first such pair differs only by their
orientation. These are genus n splittings of a closed 3-manifold that must be
stabilized at least n-2 times to become equivalent. The second is a pair of
genus n splittings of a manifold with toroidal boundary that must be stabilized
at least n-4 times to become equivalent. The last example is a pair of genus n
splittings of a closed 3-manifold that must be stabilized at least ${1/2}n -3$
times to become equivalent, regardless of their orientations. All of these
examples are splittings of manifolds that are obtained from simpler manifolds
by gluing along incompressible surfaces via "sufficiently complicated" maps.
