The weak energy condition and with the dominant energy condition allow anything from w=-1 to 1 (meaning that the magnitude of the pressure can be at most equal to the total mass-energy density). Values outside this range are needed to make wormholes and other causality violations (note that the strong energy condition is violated by dark energy). Cosmic inflation and dark energy are very close to the limit of -1, but nothing seems to come close to +1 even though it is "allowed". Both photon gases and ultra-relativistic degenerate gases are +1/3, can we do better?
Neutron stars are thought to be stiffened by a repulsion in addition to the usual degeneracy pressure. Could this repulsion stiffen the equation of state enough to exceed 1/3? It takes energy to force repelling particles together (which can be interpreted as an the energy in the virtual particles that are mediating said repulsive force). This extra energy raises the mass, behaving much like kinetic energy in degenerate matter, which would seem to prevent rising above 1/3. Is there any way around this "problem"?