The adsorption process of the single polymer chain in a polymer brush of varying surface coverages is studied by means of Monte Carlo simulations of the bond-fluctuation lattice model. Only the end monomers can adsorb at the grafting surface, whereas inner monomers interact repulsively with it. The brush builds up a steric hindrance which forces the penetrating polymer to stretch strongly and which is responsible for small adsorption probabilities at surface coverages close to the overlap density. The final step of the adsorption process is determined by a fluctuation of the end monomer around its average position, which is comparable to the initial step of the desorption process.