In recent years, many higher fullerenes that obey the isolated pentagon rule (IPR) were found capable of rearranging into molecules with adjacent pentagons and even with heptagons via chlorination-promoted skeletal transformations. However, the key fullerene, buckminsterfullerene I-h-C-60, long seemed insusceptible to such rearrangements. Now we demonstrate that buckminsterfullerene yet can be transformed by chlorination with SbCl5 at 420-440 degrees C and report X-ray structures for the thus-obtained library of non-IPR derivatives. The most remarkable of them are non-IPR C60Cl24 and C60Cl20 with fundamentally rearranged carbon skeletons featuring, respectively, four and five fused pentagon pairs (FPPs). Further high-temperature trifluoromethylation of the chlorinated mixture afforded additional non-IPR derivatives C-60(CF3)(10) and C-60(CF3)(14), both with two FPPs, and a nonclassical C-60(CF3)(15)F with a heptagon, two FPPs, and a fully fused pentagon triple. We discuss the general features of the addition patterns in the new non-IPR compounds and probable pathways of their formation via successive Stone Wales rearrangements.