We have developed a method of fabricating very shallow, gateable, undoped 2-dimensional electron gases (2DEG) and making very low resistivity contacts to these. We studied the evolution of mobility as a function of the depth of the 2DEG (from 300nm to 30nm). We extract quantitative information about the background impurity concentration in GaAs and AlGaAs, the interface roughness and the charge in the surface states. Surface charge sets an intrinsic limit to the mobility of very shallow 2DEGs. It is probably impossible, to fabricate such shallow high-mobility 2DEGs using modulation doping due to the need to accomodate a spacer layer.