First examples of multichain (polycatenar) compounds, based on the pi-conjugated benzothieno[3,2-b]benzothiophene unit are designed, synthesized, and their soft self-assembly and charge carrier mobility are investigated. These compounds, terminated by the new fan-shaped 2-brominated 3,4,5-trialkoxybenzoate moiety, form bicontinuous cubic liquid crystalline (LC) phases with helical network structure over extremely wide temperature ranges (>200 K), including ambient temperature. Compounds with short chains show an achiral cubic phase with the double network, which upon increasing the chain length, is at first replaced by a tetragonal 3D phase and then by a mirror symmetry is broken triple network cubic phase. In the networks, the capability of bypassing defects provides enhanced charge carrier mobility compared to imperfectly aligned columnar phases, and the charge transportation is non-dispersive, as only rarely observed for LC materials. At the transition to a semicrystalline helical network phase, the conductivity is further enhanced by almost one order of magnitude. In addition, a mirror symmetry broken isotropic liquid phase is formed beside the 3D phases, which upon chain elongation is removed and replaced by a hexagonal columnar LC phase.