Single-molecule electrical junctions possess a molecular core connected to source and drain electrodes via anchor groups, which feed and extract electricity from specific atoms within the core. As the distance between electrodes increases, the electrical conductance typically decreases, which is a feature shared by classical Ohmic conductors. Here we analyze the electrical conductance of cycloparaphenylene (CPP) macrocycles and demonstrate that they can exhibit a highly nonclassical increase in their electrical conductance as the distance between electrodes increases. We demonstrate that this is due to the topological nature of the de Broglie wave created by electrons injected into the macrocycle from the source. Although such topological states do not exist in isolated macrocycles, they are created when the molecule is in contact with the source. They are predicted to be a generic feature of conjugated macrocycles and open a new avenue to implementing highly nonclassical transport behavior in molecular junctions.