Is the composite of two expansive maps expansive?

We study the question whether the composite of two expansive maps is itself expansive. The answer to this question is a resounding no. We will actually establish a stronger fact. We will give an example of two dynamical systems which are topologically exact and Ruelle expanding with respect to two different, equivalent and compatible metrics, but whose composite is far from being expansive, for it is locally eventually contracting towards a fixed point a completely invariant, dense open subspace.