The chemistry initiated by C-60(x+) (with x = 1, 2, and 3) in allene and propyne has been tracked in the gas phase. Reaction rate coefficients and product distributions were measured with the Selected-Ion Flow Tube (SIFT) technique operating at 294 +/- 3 K and at a helium buffer-gas pressure of 0.35 +/- 0.01 Torr. C-60(.+) does not react with allene and propyne under SIFT conditions, but multiple addition was observed with C-60(2+). Multiple addition was also observed in the reaction of propyne with C-60(3+), but allene reacted only by electron transfer with this cation. Up to 16 molecules of allene were observed to add to C-60(2+) with a remarkable periodicity in reactivity : even-numbered adducts react approximately 10 times faster than odd-numbered adducts. Multi-collision induced dissociation (CID) experiments demonstrated that the observed multiple-adduct ions have a "ball-and-chain" structure. Mechanisms are proposed for the sequential addition reactions and for the formation of C6H7+ and C9H12+ which was seen to compete with multiple addition in the reactions of allene with C-60(2+) and of propyne with C-60(3+). The multiple-addition reactions are proposed to occur by chain propagation. The periodicity in the rates of the reactions of allene with C-60(allene)(n)(2+) is attributed to differences in charge distribution and stability associated with an alternating formation of acyclic and cyclic structures.