  Pleba\'nski's class of nonlinear vacuum electrodynamics is considered which
is for several reasons of interest at the present time. In particular the
question is answered under which circumstances Maxwell's original field
equations are recovered approximately and which post-Maxwellian effects could
arrise. To this end a weak field approximation method is developed allowing to
calculate post-Maxwellian corrections up to Nth order. In some respect this is
analogue of determining "post-newtonian" corrections from relativistic
mechanics by a low velocity approximation. As a result we got a series of
linear field equations which can be solved order by order. In this context the
solutions of the lower orders occur as source terms inside the higher order
field equations and represent a "post-Maxwellian" self-interaction of the
electromagnetic field which increases order by order. One has to decide between
problems with and without external source-terms, because without also high
frequency solutions can be approximately described by Maxwell's original
equations. The higher order approximations which describe "post-Maxwellian"
effects can give rise for experimental tests of Pleba\'nksi's class. Finally
two boundary value problems are discussed to have examples at hand.
