Weining Man,1 Aleksandar Donev,4 Frank H. Stillinger,2 Matthew T. Sullivan,1 William B. Russel,3 David Heeger,5 Souheil Inati,5 Salvatore Torquato,2,4 and P. M. Chaikin1 1Department of Physics, and Princeton Institute for the Science and Technology of Materials, Princeton University, Princeton, New Jersey 08544, USA 2Department of Chemistry, and Princeton Institute for the Science and Technology of Materials, Princeton University, Princeton, New Jersey 08544, USA 3Department of Chemical Engineering, and Princeton Institute for the Science and Technology of Materials, Princeton University, Princeton, New Jersey 08544, USA 4Program in Applied and Computational Mathematics, and Princeton Institute for the Science and Technology of Materials, Princeton University, Princeton, New Jersey 08544, USA 5Department of Psychology and Center for Neural Science, New York University, New York, New York 10003, USA Received 11 December 2004; published 19 May 2005. Phys. Rev. Lett. 94, 198001 (2005). Abstract Recent simulations indicate that ellipsoids can pack randomly more densely than spheres and, remarkably, for axes ratios near 1.25:1:0.8 can approach the densest crystal packing (fcc) of spheres, with a packing fraction of 74%. We demonstrate that such dense packings are realizable. We introduce a novel way of determining packing density for a finite sample that minimizes surface effects. We have fabricated ellipsoids and show that, in a sphere, the radial packing fraction f(r) can be obtained from V(h), the volume of added fluid to fill the sphere to height h. We also obtain f(r) from a magnetic resonance imaging scan. The measurements of the overall density favr, f(r) and the core density f0=0.74±0.005 agree with simulations.