Extreme osmotic compression of nanoemulsion droplets, achieved by ultracentrifugation, can create solidlike biliquid foams without causing significant droplet coalescence. Using small-angle neutron scattering ( SANS), we probe the structure of a uniform silicone oil-in-water nanoemulsion stabilized by sodium dodecyl sulfate over a wide range of volume fractions, phi, up to and beyond the limit associated with maximal random jamming of spheres, phi(MRJ)=0.64. Although some features in the structure can be understood at lower phi using simple predictions for hard spheres, the anionic repulsion and deformability of the droplet interfaces creates departures from these predictions at higher phi. For phi near and beyond phi(MRJ), the effective structure factor, S-eff, as a function of wavenumber, q, exhibits a primary peak that is subunity. We speculate that this striking feature is due to the deformation of the droplets into nonspherical shapes as the system begins to approach the limit of a random array of nanoscopic thin films, Plateau borders, and vertexes characteristic of a polyhedral foam.