Several years ago, it was proposed that a dense nonneutral plasma could be produced in a Penning trap. Nonneutral plasmas have excellent confinement. Thus, such a dense plasma might produce simultaneously high density and good confinement (as needed for fusion). Recently, this theoretical conjecture has been demonstrated in a small (3 mm radius) electron experiment (PFX). Densities up to 35 times the Brillouin density (limiting number density in a static trap) have been inferred from the observed strong (100:1) spherical focussing. Electrons are injected at low energy from a single pole of the sphere. A surprising observation is the self-organization of the system into a spherical state, which occurs precisely when the trap parameters are adjusted to produce a spherical well. This organization is observed by a bootstrapping which produces a hysteresis. Additional observations which confirm the dense spherical focus are energy-scattered electrons and deflections of an electron probe beam by the space charge of the central focus.