Tetrahedra may be the ultimate frustrating, disordered glass forming units. Our experiments on tetrahedral dice indicate the densest (volume fraction phi=0.76+/-.02, compared with phi(sphere)=0.64), most disordered, experimental, random packing of any set of congruent convex objects to date. Analysis of MRI scans yield translational and orientational correlation functions which decay as soon as particles do not touch, much more rapidly than the approximately 6 diameters for sphere correlations to decay. Although there are only 6.3+/-.5 touching neighbors on average, face-face and edge-face contacts provide enough additional constraints, 12+/-1.6 total, to roughly bring the structure to the isostatic limit for frictionless particles. Randomly jammed tetrahedra form a dense rigid highly uncorrelated material.