Traditionally, high strength lightweight materials are created from mechanically rigid structures with a combination of compressive and tensile forces. By contrast, purely compressive materials such as sand and gravel do not typically offer a high strength to weight ratio. By considering sphere packings in which all forces are compressive, we determine the limits on creating low density rigid systems. An Apollonian sphere packing proves that a rigid packing can completely fill space, but what is the lowest density a stable packing can achieve? The previously known lowest density packings are constructed by diluting simple crystals. We present a new construction based on rigid quasi-one dimensional structures. These constructions can be used to create rigid packings with densities arbitrarily close to zero. We demonstrate the rigidity of these low density packings using new procedures and we explore the properties of these configurations to gain a deeper understanding of the limits of rigidity in repulsive systems. Such constructions not only lay an old puzzle to rest, but enable the development of new lightweight materials.