How accurate is pair additivity in describing interactions between soft polymer-based nanoparticles? Using numerical simulations we compute the free energy cost required to overlap multiple chains in the same region of space, and provide a quantitative measure of the effectiveness of pair additivity as a function of chain number and length. Our data suggest that pair additivity can indeed become quite inadequate as the chain density in the overlapping region increases. We also show that even a scaling theory based on polymer confinement can only partially account for the complexity of the problem. In fact, we unveil and characterize an isotropic to star-polymer crossover taking place for large number of chains, and propose a revised scaling theory that better captures the physics of the problem.