Is there any upper bound for the refractive index of a theoretical material to occur? Would the relativistic effects be taken into consideration?
There are some clever techniques where by using a strong laser beam to modify the properties of a gas or solid, one can make the refractive index for a second 'probe' beam to be as high as you like, over a narrow range of frequencies.
To learn more, look up the phenomenon called 'slow light'. However, in such cases it becomes debatable whether the simple name 'light' is the best name for the disturbance which propagates through the material. Whenever light propagates through a material with index differing from the value 1, the material itself has internal oscillations (e.g. oscillating dipoles) and what propagates is a joint oscillation of the electromagnetic field along with these material oscillations. In the case of 'slow light' this is especially true.
In the limit where the index goes to infinity, the 'light' ceases to propagate. What happens is that the oscillation has passed totally to the medium, and it can be frozen there, like a form of memory. Then by adjusting the 'pump' beam the index can be brought to a lower value again and the frozen disturbance propagates once more.