The laser desorption of Ck (k = 60 and 70) fullerene is known to produce a broad distribution of cluster sizes strongly peaking around the integer multiples of original fullerene mass. The "exact dimers" (Cn clusters with n = 2k) and species with slightly fewer atoms (n even) have been characterized previously as fully coalesced large single-shell fullerenes and [2 + 2] cycloadducts. Presently, we investigate the species encountered on the high-mass sides of exact dimers, that is, the clusters with n > 2k (n even), using high-resolution ion mobility measurements. Specifically, the drift time distributions for Cn+ and Cn- with n = 122-128,132-136, and 142-146 have been obtained and compared with the results of trajectory calculations for various trial geometries optimized using the density functional tight binding and semiempirical (AMI) calculations. We find that, besides the normal near-spherical fullerenes and [2 + 2] cycloadducts, these species assume the "ball-and-chain" structures consisting of two fullerene cages (not necessarily those of the original material) connected by chains up to eight atoms long. C122 and C132 cations and anions also reveal a substantial abundance of isomers where the C-C unit is sandwiched between the two fullerenes. Taken in conjunction with earlier findings for smaller fullerene dimers, presently reported results have allowed us to develop a comprehensive model for the chemical reactions occurring in the hot fullerene plasma.