We consider the growth of a polymer layer on a fiat surface in a good solvent by in situ polymerization. This is viewed as a modified form of diffusion-limited aggregation without branching. We predict theoretically the formation of a pseudo-brush with density phi(z) proportional to z(-2/3) and characteristic height H proportional to t(3). These results are found by combining a mean-field treatment of the diffusive growth (marginally valid in three dimensions) with a scaling theory (Flory exponent nu = 3/5) of the growing polymers. We confirm their validity by Monte Carlo simulations.