Because the commonly adopted viewpoint that the Keldysh parameter gamma determines the dynamical regime in strong field physics has long been demonstrated to be misleading, one can ask what happens as relevant physical parameters, such as laser intensity and frequency, are varied while gamma is kept fixed. We present results from our one-and fully three-dimensional quantum simulations of high-order-harmonic generation (HHG) from various bound states of hydrogen with n up to 40, where the laser intensities and the frequencies are scaled from those for n = 1 in order to maintain a fixed Keldysh parameter gamma < 1 for all n. We find that as we increase n while keeping gamma fixed, the position of the cutoff scales in a well-defined manner. Moreover, a secondary plateau forms with a new cutoff, splitting the HHG plateau into two regions. The first of these subplateaus is composed of lower harmonics, and has a higher yield than the second one. The latter extends up to the semiclassical I-p + 3.17U(p) cutoff. We find that this structure is universal, and the HHG spectra look the same for all n greater than or similar to 10 when plotted as a function of the scaled harmonic order. We investigate the n, l, and momentum distributions to elucidate the physical mechanism leading to this universal structure.