Swelling initially solvent-free monolayers of anchored polymer chains with a gradually increasing amount of solvent significantly influences their thermodynamics, especially the lateral pressure. Upon increasing the amount of solvent, the lateral pressure initially increases inversely proportional to the square of the average volume fraction of the anchored chains but finally levels off to a limiting value independent of the composition. With increasing area per anchored chain the limiting value of the lateral pressure decreases, while the amount of solvent needed to reach the limiting value increases. The observed effects are in good agreement with predictions based on analytical mean field theory.