The reactions of the fullerene ions C60.+, C60(2+), and C-60(.3+) with the neutrals H2O, CH3OH, CH3CH2OH, CH3CH2CH2OH, (CH3)2CHOH, CH3OCH3, (CH3CH2)2O, and c-C4H8O in helium at 0.35 +/- 0.01 Torr and 294 +/- 2 K have been studied using a selected-ion flow tube. Association was the most commonly encountered primary product channel seen in the reactions of C60(2+): in keeping with earlier studies, there was a clear dependence of the efficiency of association (and of reactivity in general) upon the size of the neutral. Other product channels evident in the reactions of the dication were charge transfer (the major product channel seen in the reaction with diethyl ether) and hydroxide abstraction to form the ion C60OH+ (in the reactions with ethanol and 2-propanol). Charge transfer and hydroxide abstraction were seen in several reactions of the trication, C-60(.3+): association was observed as a minor channel in the reactions with methanol, ethanol, and 1 -propanol. A clear difference was observed in the reactivity of the polycationic adducts of alcohols and ethers: alcohol adducts were observed to react further by efficient proton transfer to the parent alcohol, whereas the adducts of ethers did not display subsequent proton transfer to the parent ether. This difference in reactivity is interpreted in terms of the difference in ease of proton loss from the structures ascribed to the fullerene polycation adducts of alcohols and ethers. The monocation C60.+ was unreactive with all of the species studied here: monocationic product ions (C60OH+, C60OR+) were also observed to be unreactive with the neutrals from which they were produced. The implications of the non-reactivity of C60.+ and the reactivity of C-60(2+) for the chemical evolution of interstellar clouds and circumstellar shells are briefly discussed.