Non-additivity of energy (thermodynamics) In page 15, Chapter 1 (Laws of Thermodynamics) of the book "An Introduction to the Study of Stellar Structure" by Nobel laureate S. Chandrasekhar, it is mentioned (in context of total energy of two subsystems brought in contact):

In general when two bodies are brought in contact, the energy is not additive; it is easy
  to see however, that the deviation is directly proportional to common surface area of the bodies, hence, for the large volumes the deviations from the additive law can be neglected

I always thought that energy is additive as a fundamental property (or by definition). I did not find anything further to understand the statement. Please, help me understand this.
 A: It hinges on the sentences you did not quote, and on whether one is talking of an adiabatic system or not. He  assumes two systems adiabatically isolated then their internal energy can be added. Then when brought in contact the adiabatic isolation no longer holds, potential energies and radiations exchanged are not additive in a simple manner thermodynamially.
The view is that two bodies separated by a permeable to matter surface cannot be in an adiabatic state:

According to Max Born, the transfer of matter and energy across an open connection "cannot be reduced to mechanics". In contrast to the case of closed systems, for open systems, in the presence of diffusion, there is no unconstrained and unconditional physical distinction between convective transfer of internal energy by bulk flow of matter, the transfer of internal energy without transfer of matter (usually called heat conduction and work transfer), and change of various potential energies.

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In particular, between two otherwise isolated open systems an adiabatic wall is by definition impossible.

They solve the problem by going to the law of  conservation of energy which always holds whether the system can be characterized as adiabatic or isolated.
Chandrasekhar needed a useful approximation for his astrophysical arguments and assumes that the contact surface  is small for large bodies and he can ignore the potential radiation etc energies that have to rebalance  when joining two separate with a large contact surface.
A: He is talking about two systems in adiabatic inclosures (see p 12). These are inclosures where heat cannot penetrate the walls and the only way to change the internal energy U of the system is to do work on it by displacing a wall. 
U is not the total energy of the system. There is also heat, Q. 
The only other case he considers is where a wall is diathermic. Heat can penetrate this kind of wall. 
When two systems are brought into contact, he says the change in internal energy is proportional to the area of common wall. He must mean that some part of each system's wall becomes a movable common wall, so the systems can do work on each other. 
If the common wall is diathermic, the total energy, Q + U, is additive. But internal energy, U, is not. 
