  We show that a strong form of the so called Lindstrom's Theorem fails to
generalize to extensions of L_{kappa,omega} and L_{kappa,kappa}: For weakly
compact kappa there is no strongest extension of L_{kappa,omega} with the
(kappa,kappa)-compactness property and the Lowenheim-Skolem theorem down to
kappa. With an additional set-theoretic assumption, there is no strongest
extension of L_{kappa,kappa} with the (kappa,kappa)-compactness property and
the Lowenheim-Skolem theorem down to <kappa.
