Hybrid organic-inorganic perovskites, such as methylammonium lead tri-iodide (), are interesting candidates for efficient absorber materials in next-generation solar cells, partly due to an unusual combination of low exciton-binding energy and strong optical absorption. Excitonic effects in this material have been subject to debate both for experiment and theory, indicating a need for better understanding of the screening mechanisms that act upon the electron-hole interaction. Here, we use cutting-edge first-principles theoretical spectroscopy, based on density-functional and many-body perturbation theory, to study atomic geometries, electronic structure, and optical properties of three polymorphs and find good agreement with earlier results and experiment. We then study the influence of free electrons on the electron-hole interaction and show that this explains consistently smaller exciton-binding energies, compared to those in the material without free electrons. Interestingly, we also find that the absorption line shape strongly resembles that of the spectrum without free electrons up to high free-electron concentrations. We explain this unexpected behavior by formation of Mahan excitons that dominate the absorption edge, making robust against free-electron-induced changes observed in other semiconductors.