Density of randomly packing a box

I want to throw a lot of copies of an object of nonzero volume, randomly into a large box. Ignoring boundary effects of the box, with which type of object will the expected packing density be the largest?

Is anything known about the 2D analogue, with any kind of random dropping of objects.

The beautiful paper by Torquato and Stillinger, "Jammed hard-particle packings: From Kepler to Bernal and beyond" (Rev. Mod. Phys. 82, 2633–2672 (2010)), will answer many of your questions. It focuses on spheres, but also touches upon ellipsoids, superballs, polyhedra, higher dimensions, and non-Euclidean geometries. It is an up-to-date survey with pages of references. You may recognize Torquato as a primary mover in the recent spectacular advances on tetrahedral packing.

I think that your question, made precise in some way, is likely open. But there are octahedral packings of density 0.947, considerably higher than the best tetrahedral or sphere packings. So a regular octahedron is a candidate answer.

Incidentally, this paper says that the notion of "random close packing" is now passé, being displaced by "maximally random jamming," a more precise concept.

Some specific information on random sphere packings can be found in my MO question "Average degree of contact graph for balls in a box."