The structure of graphene grown in chemical vapor deposition (CVD) is sensitive to the growth condition, particularly the substrate. The conventional growth of high-quality graphene via the Cu-catalyzed cracking of hydrocarbon species has been extensively studied; however, the direct growth on noncatalytic substrates, for practical applications of graphene such as current Si technologies, remains unexplored. In this study, nanocrystalline graphene (nc-G) spirals are produced on noncatalytic substrates by inductively coupled plasma CVD. The enhanced out-of-plane electrical conductivity is achieved by a spiral-driven continuous current pathway from bottom to top layer. Furthermore, some neighboring nc-G spirals exhibit a homogeneous electrical conductance, which is not common for stacked graphene structure. Klein-edge structure developed at the edge of nc-Gs, which can easily form covalent bonding, is thought to be responsible for the uniform conductance of nc-G aggregates. These results have important implications for practical applications of graphene with vertical conductivity realized through spiral structure.