In this article, we consider an anisotropic finite-range bond percolation model on Z(2). On each horizontal layer {(x, i): x is an element of Z} we have edges <(x, i), (y, i)> for 1 <= vertical bar x-y vertical bar <= N. There are also vertical edges connecting two nearest neighbor vertices on distinct layers <(x, i), (x, i+1)> for x, i is an element of Z. On this graph we consider the following anisotropic independent percolation model: horizontal edges are open with probability 1/(2N), while vertical edges are open with probability epsilon to be suitably tuned as N grows to infinity. The main result tells that if epsilon = kappa N-2/5, we see a phase transition in kappa: positive and finite constants C-1, C-2 exist so that there is no percolation if kappa < C-1 while percolation occurs for kappa > C-2. The question is motivated by a result on the analogous layered ferromagnetic Ising model at mean field critical temperature [11, J. Stat. Phys. 161, (2015), 91-123] for which the authors showed the existence of multiple Gibbs measures for a fixed value of the vertical interaction and conjectured a change of behavior in kappa when the vertical interaction suitably vanishes as kappa gamma(b), where 1/gamma is the range of the horizontal interaction. For the product percolation model we have a value of b that differs from what was conjectured in that paper. The proof relies on the analysis of the scaling limit of the critical branching random walk that dominates the growth process restricted to each horizontal layer and a careful analysis of the true horizontal growth process, which is interesting by itself. This is inspired by works on the long range contact process [17, Probab. Th. Rel. Fields 102, (1995), 519-545]. A renormalization scheme is used for the percolative regime.