Constraining the planetary composition is essential for exoplanetary characterization. In this paper, we use a statistical analysis to determine the characteristic maximum (threshold) radii for various compositions for exoplanets with masses up to 25 Earth masses ($M_{\oplus}$). We confirm that most planets with radii larger than 1.6 Earth radii ($R_{\oplus}$) are not rocky, and must consist of lighter elements, as found by previous studies. We find that planets with radii above 2.6 $R_{\oplus}$ cannot be pure-water worlds, and must contain significant amounts of hydrogen and helium (H–He). We find that planets with radii larger than about 3 R_{\oplus}$, 3.6 $R_{\oplus}$, and 4.3 $R_{\oplus}$ are expected to consist of 2%, 5%, and 10% of H–He, respectively. We investigate the sensitivity of the results to the assumed internal structure, the planetary temperature and albedo, and the accuracy of the determination of mass and radius. We show that the envelope’s metallicity, the percentage of H–He, and the distribution of the elements play a significant role in the determination of the threshold radius. Finally, we conclude that, despite the degenerate nature of the problem, it is possible to put limits on the possible range of compositions for planets with well-measured mass and radius.