Is action an extensive quantity - always? Action, the integral over time of the kinetic minus the potential energy seems to be an extensive quantity. (There is nothing serious coming up in Google on this issue. Neither on Google Scholar.)
In classical mechanics and in special relativity, action is proportional to mass; so it is extensive. Also electrodynamics seems to confirm this.
Is action also extensive in quantum theory? And in general relativity?
 A: Principal property of the action is that the motion of the particle in its entirety covers a certain amount of phase space which corresponds to the action attributed to the particle. This phase space volume depends above all on the initial condition of the particle $(x_0,y_0,z_0,p_{x0}, p_{y0}, p_{z0})$.
The concept of "extensive" is used in thermodynamics and statistical mechanics, therefore it can only be applied to an ensemble of particles.
If an ensemble of particles is considered (only then extensivity makes sense) each individual particle of the ensemble has a particular action which can be small or large depending on its initial conditions.
Under these circumstances an action average value of particle ensemble can be defined, which in beam physics is also known as emittance.
However upon increasing the number of particles, the emittance does not necessarily change its value, so it cannot be considered as genuinely extensive. You can inject in a beam within an accelerator 10mA ($N$ particles) or 100mA ($10\times N$ particles) , the emittance remains constant -- (if not an instability is hit).
In quantum physics if an ensemble of fermions is considered they need a minimum amount of phase space, so if the number of fermions is increased probably one needs the double amount of phase space volume for them. So for quantum fermions one could say that the average value of action of the considered fermion ensemble is extensive.
However, for an ensemble of bosons no minimum phase space volume is required. One could put as much bosons as wanted into a certain phase space volume. Therefore for quantum bosons one could say that the average value of action of the boson ensemble is not extensive at all.
