We present a simple algorithm that produces a so called growing Ramified Random Apollonian Networks (R-RAN) in any dimension $d$. Analytical derivations show that these networks still exhibit small-word and scale-free characteristics. To characterize further the structure of R-RAN, we introduce new parameters that we refer to as the parallel degree and the ramification coefficient, that determine locally and in average the number of vertices inside the (d+1)-cliques composing the network. We provide analytical derivations for the computation of the degree and parallel degree distributions, ramification and clustering coefficients. We give an upper bound for the average path lengths for R-RAN and finally show that our derivations are in very good agreement with our simulations.