How to count microstates? In statistical mechanics the entropy of a macroscopic system in equilibrium, assuming equipartition, is equal to the logarithm of the number of microstate compatible with it, up to a fixed and well-defined constant. This coincides with the traditional definition of the entropy (which is where the constant comes from).
This definition should be independent of the underlying theory, but how is that possible? When substructure is considered or discovered (e.g. of particles in a gas), or when theories are replaced or refined, the number of microstates (according to the theory) will generally change. It doesn't seem to be the case however that the absolute entropy can not be computed unless a complete theory of every detail of the system is known. Moreover entropy is something that can be measured without having any idea of theory or substructure.
Finally many or most systems have an infinite state space, for example a classical many-particle macrostate will correspond to an infinite subset of the phase space, and an ideal quantum gas at sufficiently high temperature/energy so that unbounded states are accessible will correspond to an infinite dimensional subspace of its Hilbert space. How would this (i.e. the counting of states) work there? 
Without a doubt this is a very naive question, but i really have no idea of what the answer could be.
 A: If we describe a system using quantum mechanics and then consider the classical regime where we can give an approximate description using a classical phase phase, then what you find is that the density of states per unit phase space volume per particle $d^3xd^3p$ is $\frac{1}{h^3}$. Planck's constant enters in here, which you can interpret as a scaling constant. So, you can indeed choose to "zoom out" so that the phase space description become better and better and everything looks more classical, but the density of states per unit phase spaced volume in your more coarse grained phase space then will increase.
While we can do "classical statistical physics" where we start with a classical phase phase, this is physically not the right way to do statistical physics. Given some well defined physical system you could only have rescaled the coordinates and momenta by some finite amount, so that in terms of the new coordinates the density of states per particle would become $\frac{A}{h^3}$ where $A$ is the Jacobian of the scaling transform. The idea of being allowed to start at the infinite scaling limit and then not having to pay a price for that (like not being able to define the absolute entropy) is flawed. 
