Given a "target" sphere of radius r(1) and "probe" spheres of radius r(2), we consider, as a function of r(2)/r(1), how many probe spheres, on average, can be attached to the target sphere if (1) the attachment sites are chosen at random, (2) the probe spheres are not permitted to overlap, and (3) each attachment is irreversible. We also consider two separate extremes for selecting new attachment sites : Each probe sphere is either permitted to diffuse into place from a large distance, or the attachment site is chosen completely at random. Diffusion-controlled attachment produces a slightly higher packing density than completely random attachment.