Is it possible to generate electricity by using two inert electrodes in a galvanic cell?
A non-chemical way to generate electricity using two inert electrodes in salt water is through magnetohydrodynamics. If the salt water moves past the electrodes in a magnetic field, it generates an electric current without consuming the electrodes or electrolyte. 

This could make an interesting "magic trick" or puzzle: use a moving magnetic field, e.g. a magnet attached to a motor, in a box below a beaker filled with salt water, and insert two carbon electrodes connected to an AC current meter. Of course, make sure the motor is very quiet.