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The recent discovery of superconductivity at the interface of two non-superconducting materials has received much attention. In cuprate bilayers, the critical temperature (Tc) can be significantly enhanced compared with single-phase samples. Several explanations have been proposed, invoking Sr interdiffusion, accumulation and depletion of mobile charge carriers, elongation of the copper-to-apical-oxygen bond length, or a beneficial crosstalk between a material with a high pairing energy and another with a large phase stiffness. From each of these models, one would predict Tc to depend strongly on the carrier density in the constituent materials. Here, we study combinatorial libraries of La(2-x)Sr(x)CuO4-La2CuO4 bilayer samples--an unprecedentedly large set of more than 800 different compositions. The doping level x spans a wide range, 0.15 < x < 0.47, and the measured Hall coefficient varies by one order of magnitude. Nevertheless, across the entire sample set, Tc stays essentially constant at about 40 K. We infer that doping up to the optimum level does not shift the chemical potential, unlike in ordinary Fermi liquids. This result poses a new challenge to theory--cuprate superconductors have not run out of surprises.