  We predict the existence of exchange broadening of optical lineshapes in
disordered molecular aggregates and a nonuniversal disorder scaling of the
localization characteristics of the collective electronic excitations
(excitons). These phenomena occur for heavy-tailed L\'evy disorder
distributions with divergent second moments - distributions that play a role in
many branches of physics. Our results sharply contrast with aggregate models
commonly analyzed, where the second moment is finite. They bear a relevance for
other types of collective excitations as well.
