An unusual growth phenomenon, with no precedent in vapor-phase thin film growth, is described here, for the case of the growth of stacked multiple layers of vertically aligned carbon nanotubes(1-6) on solid substrates. As multiple layers of ordered nanotubes are sequentially deposited from the vapor onto the substrate, each layer nucleates and grows from the original substrate surface at the bottom of the existing multiple stacks of nanotubes. In contrast to conventional understanding of thin film deposition,(7) the mechanism here has similarities to porous oxide film formation on surfaces.(8) The stacked layers of aligned nanotubes act as fully permeable membranes for the downward diffusion of growth precursor vapors, allowing growth to occur at the buried solid interface. The preexisting multiple nanotube stacks lift up to accommodate the vertical growth of fresh layers, allowing the formation of nanotube towers extending in millimeter lengths. Our results provide evidence for a new growth phenomenon, characterized by selective, interface-driven, bottom-up growth of self-assembled nanowires at buried interfaces, covered with weakly adhering thick porous membranes.