Fractal worlds with limited connectivity are the topological result of growing graphs from chaotic series. We show how this model presents original characteristics which cannot be detected by means of the standard network descriptors. In detail, intrinsic inaccessibility to the fully connected configuration is demonstrated to be a universal feature associated with this family of graphs and strictly related to the fractality of a specific ``chaotic source''. Here we discuss the potential of our model to be a generator of fractal graphs and also a self-consistent tool for differentiating chaotic dynamics from stochastic processes.