  High precision measurements of the Casimir effect and recent applications to
micro electromechanical systems raise the question of how large the Casimir
force can be made in an arbitrarily small device. Using a simple model for the
metal boundary in which the metal is perfectly conducting at frequencies below
plasma frequency omega_p and perfectly transparent above such frequency, I find
that the Casimir force for plate separations a<lambda_p/2, where lambda_p is
the plasma wavelength is given by -(h omega_p^4)/(24 pi^2 c^3) which is
independent of a. This result is considered the maximum value of the Casimir
force for non-ideal metallic boundaries as calculated by quantum field theory.
It differs from predictions of non retarded Van der Waals theory. Implications
of this result for geometries different from the planar one and in particular
for the hollow metallic sphere are discussed.
