The topological p-wave pairing of composite fermions, believed to be responsible for the 5/2 fractional quantum Hall effect (FQHE), has generated much exciting physics. Motivated by the parton theory of the FQHE, we consider the possibility of a new kind of emergent "superconductivity" in the 1/3 FQHE, which involves condensation of clusters of n composite bosons. From a microscopic perspective, the state is described by the n (n) over bar 111 parton wave function P-LLL Phi(n)Phi(n)*Phi(3)(1), where Phi(n) is the wave function of the integer quantum Hall state with n filled Landau levels and P-LLL is the lowest-Landau-level projection operator. It represents a Z(n) superconductor of composite bosons, because the factor Phi(3)(1) similar to Pi(j111 and 3 (3) over bar 111 states are at least as plausible as the Laughlin wave function for the exact Coulomb ground state at filling nu = 7/3, suggesting that this physics is possibly relevant for the 7/3 FQHE. The Z(n) order leads to several observable consequences, including quasiparticles with fractionally quantized charges of magnitude e/(3n) and the existence of multiple neutral collective modes. It is interesting that the FQHE may be a promising venue for the realization of exotic Z(n) superconductivity.