  Discrete-spin systems with maximally random nearest-neighbor interactions
that can be symmetric or asymmetric, ferromagnetic or antiferromagnetic,
including off-diagonal disorder, are studied, for the number of states $q=3,4$
in $d$ dimensions. We use renormalization-group theory that is exact for
hierarchical lattices and approximate (Migdal-Kadanoff) for hypercubic
lattices. For all d>1 and all non-infinite temperatures, the system eventually
renormalizes to a random single state, thus signaling qxq degenerate ordering.
Note that this is the maximally degenerate ordering. For high-temperature
initial conditions, the system crosses over to this highly degenerate ordering
only after spending many renormalization-group iterations near the disordered
(infinite-temperature) fixed point. Thus, a temperature range of short-range
disorder in the presence of long-range order is identified, as previously seen
in underfrustrated Ising spin-glass systems. The entropy is calculated for all
temperatures, behaves similarly for ferromagnetic and antiferromagnetic
interactions, and shows a derivative maximum at the short-range disordering
temperature. With a sharp immediate contrast of infinitesimally higher
dimension 1+\epsilon, the system is as expected disordered at all temperatures
for d=1.
