A renormalization group approach to self-similar multilayer aggregation processes is suggested on the basis of the following assumptions. 1. The aggregate is generated by a layer-by-layer process, the rate of growth of a given layer being proportional to the number of free sites in the preceding layer. 2. The process is self-similar, that is, from layer to layer the rates of growth are scaled by a subunitary positive Grossmann factor a. It is shown that this mechanism of growth leads to temporal scale invariance. For large times the total rate of the process decreases in time according to a hyperbolic law modulated by oscillations occurring on a logarithmic time scale, with a period equal to the logarithm of the reciprocal value of the scaling factor a. In the long run the time-dependence of the total mass of the aggregate is given by a logarithmic function of time also modulated by logarithmic oscillations. The model is applied to the tunnel-assisted wet oxidation of silicon. It is assumed that multiple layers of silicon oxide are generated by a mechanism that involves a tunneling process through the silicon oxide layers. In this case Grossmann's scaling factor and the period of logarithmic oscillations have simple physical interpretations: the scaling factor is the transparency of a layer of silicon oxide, and the period of logarithmic oscillations is a measure of the tunneling length. The logarithmic oscillations of the model give a theoretical description of the stepped behavior of the oxidation process observed in the experiments reported in the literature. The presented results are of interest both from the points of view of statistical physics of fractals and nonlinear chemical kinetics.