Do reversible processes exist in Nature? Classical thermodynamics always discusses entropy in the light of reversible processes, and it lies at the heart of the definition of entropy. But do these reversible processes exist in Nature, or are they just a gedankenexperiment?
Since if they do not exist, then classical thermodynamics gives a lower bound on the increase of entropy, but says nothing about the difference in entropy between the reversible and irreversible process. Doesn't this make the classical definition useless for a quantitative approach? Or is there proof that the entropy difference between the reversible and irreversible process is small for most everyday cases?  
 A: Of course, thermodynamics says things about differences in entropy between reversible and irreversible processes. In fact, we can analyse how irreversible a process was and what to do in order to, for instance, extract more work from it. That's a very important part of thermodynamics.
But, no, there are no truly reversible processes — at least for daily macroscopic, bulk phenomena; I might be unaware of some crazy quantum-relativistic-boson-fermion experiment :). In practice, all real processes are irreversible, even though some can be good approximations of reversible ones (e.g. if you go really slowly).
Also, your statement "classical thermodynamics gives a lower bound on the increase of entropy" is a bit weird. That "lower bound" would be zero, so I'd prefer to say that it states that global entropy never decreases. Apart from that, classical thermodynamics also goes on to explore quantities like availability, irreversibility and efficiency, and it can do a good work of optimizing the hell out of your irreversible processes.
