  We consider the relation between higher spin gauge fields and real Kac-Moody
Lie algebras. These algebras are obtained by double and triple extensions of
real forms g_0 of the finite-dimensional simple algebras g arising in
dimensional reductions of gravity and supergravity theories. Besides providing
an exhaustive list of all such algebras, together with their associated
involutions and restricted root diagrams, we are able to prove general
properties of their spectrum of generators with respect to a decomposition of
the triple extension of g_0 under its gravity subalgebra gl(D,R). These results
are then combined with known consistent models of higher spin gauge theory to
prove that all but finitely many generators correspond to non-propagating
fields and there are no higher spin fields contained in the Kac-Moody algebra.
