  In this article we prove local well-posedness of the system of equations
$\partial_t h_{i}= \sum_{j=1}^{i}\partial_x^2 h_{i}+ (\partial_x h_{i})^2 + \xi
$ on the circle where $1\leq i\leq N$ and $\xi$ is a space-time white noise. We
attempt to generalize the renormalization procedure which gives the Hopf-Cole
solution for the single layer equation and our $h_1$ (solution to the first
layer) coincides with this solution. However, we observe that cancellation of
logarithmic divergences that occurs at the first layer does not hold at higher
layers and develop explicit combinatorial formulae for them.
