Question

Why oxygen is not combustible?

Answer 1

To add some detail to the other answers to explain this lets look at the definition of combustion:

Combustion: rapid chemical combination of a substance with oxygen, involving the production of heat and light.

A combustible substance is a substance which undergoes combustion, and and explosive:

A chemical or compound that causes a sudden, almost instantaneous release or pressure, gas, heat and light when subjected to sudden shock, pressure, high temperature or applied potential.

There is really only one reaction that oxygen undergoes in combination with oxygen or when subjected to high temperatures, electrical potentials or similar:

3O2 -> 2O3 (In other words the formation of ozone)

However this is an endothermic reaction, which means that it consumes energy from it's environment rather than releasing energy to it's environment therefore this process is neither combustion nor is it an explosion. In fact if you could produce sufficient amounts of ozone and somehow prevent the rapid breakdown of the unstable product the result would be more the exact opposite with a drop in temperature, drop in pressure and an implosion instead. Of course this does not actually happen as ozone is too unstable to hang around in significant quantities unless being constantly replenished by the input of energy.