Summary: A numerical investigation is presented for the peak sidelobe level (PSL) of Legendre sequences, maximal length shift register sequences (\(m\)-sequences), and Rudin-Shapiro sequences. The PSL gives an alternative to the merit factor for measuring the collective smallness of the aperiodic autocorrelations of a binary sequence. The growth of the PSL of these infinite families of binary sequences is tested against the desired growth rate \(o(\sqrt{n\ln n})\) for sequence length \(n\). The claim that the PSL of \(m\)-sequences grows like \(O(\sqrt n)\), which appears frequently in the radar literature, is concluded to be unproven and not currently supported by data. Notable similarities are uncovered between the PSL and merit factor behavior under cyclic rotations of the sequences.