Exploratory calculations using accretionary theory are made to demonstrate plausible sizes of second-largest, third-largest, etc., bodies at the close of planet formation in heliocentric orbits near the planets, assuming asteroid-like size distributions at the start of the calculation. Many satellite-sized bodies are found to be available for capture, cratering, or collisional fragmentation. In the case of Earth-sized planets, the models suggest second-largest bodies of 500 to 3000 km radius, and tens of bodies larger than 100 km radius. Many of these interact with the planet before suffering any fragmentation events with each other. Collision of a large body with Earth could eject iron-deficient crust and upper mantle material, forming a cloud of refractory, volatile-poor dust that could form the Moon. Other satellite systems may have been affected by major capture or collision events of chance character.