Highly conductive single-molecule junctions typically involve pi-conjugated molecular bridges, whose frontier molecular orbital energy levels can be fine-tuned to best match the Fermi level of the leads. Fully saturated wires, e.g., alkanes, are typically thought of as insulating rather than highly conductive. However, in this work, we demonstrate in silico that significant zero-bias conductance can be achieved in such systems by means of topology. Specifically, caged saturated hydrocarbons offering multiple sigma-conductance channels afford transmission far beyond what could be expected based upon conventional superposition laws, particularly if these pathways are composed entirely from quaternary carbon atoms. Computed conductance of molecular bridges based on carbon nanothreads, e.g., polytwistane, is not only of appreciable magnitude; it also shows a very slow decay with increasing nanogap, similarly to the case of pi-conjugated wires. These findings offer a way to manipulate the transport properties of molecular systems by means of their topology, alternatively to the traditionally invoked electronic structure.