From a statistical mechanics point of view, the extensivity of thermodynamics arises from the fact that we can neglect interaction terms in the Hamiltonian. This allows us to infer that when we put two systems in contact, the total energy is approximately equal to the sum of the individual systems' energy. If the systems are in thermal equilibrium it follows that the entropy is extensive as well.
However, it is well known that if the interaction terms cannot be neglected then these extensivity results do not follow. In theory, this should typically be the case for very small systems. (It should also be the case for systems with very long range interactions, such as gravitational systems, but for this question I'm more interested in systems that are made of atoms and interact electromagnetically.)
I'm interested in whether there has been any experimental work designed to measure this effect experimentally. In principle it could be done with calorimetry, but I'm not sure how sensitive the measurements would need to be. There might be ways to multiply the effect into a larger one, such as letting a very large number of tiny particles come into contact, rather than just two. Indeed, it might be a well-known effect that just isn't usually explained in these terms.
I'd be particularly interested in work that measures the non-extensivity of both internal energy and entropy, in the same system, using independent measurements. However, any empirical work that addresses this issue is of interest.
From searching Google I did come across a few experimental papers that mention non-extensivity. However, they seem to be concerned with testing the predictions of Tsallis entropy maximisation, which isn't what I'm interested in here. I'm really looking for calorimetric measurements of the non-additivity of the Hamiltonian in small systems, and the resulting changes in thermodynamic properties.