What are the possible changes that might occur if I add an inert gas to a chamber containing another few different gases?
Quoting from the Wikipedia page on Le Chatelier's principle, we have to distinguish two cases:

The total volume is constant:

Adding an inert gas into a gas-phase equilibrium  does not result in a shift of the equilibrium in this case. This is due to the fact that adding an inert gas does not change the partial pressures of the other gases in the container. While it is true that the total pressure of the system increases, the total pressure does not have any effect on the equilibrium constant; rather, it is a change in partial pressures that will cause a shift in the equilibrium.


The total volume is not constant:

If, however, the volume is increased, the partial pressures of all gases will decrease which results in a shift of the equilibrium towards the side with the greater number of moles of gas.