  We present studies of high-order harmonic generation (HHG) at laser
intensities well above saturation. We use driving laser pulses which present a
particular electron dynamics in the turn-on stage. Our results predict an
increasing on the harmonic yield, after an initial dropping, when the laser
intensity is increased. This fact contradicts the general belief of a
progressive degradation of the harmonic emission at ultrahigh intensities. We
have identified a particular set of trajectories which emerges in the turn-on
stage of these singular laser pulses, responsible of the unexpected growth on
the harmonic efficiency at this high intensity regime. Our study combines two
complementary approaches: classical analysis and full quantum mechanical
calculations resulting from the numerical integration of the 3-dimensional
time-dependent Schr\"odinger equation complemented with the time-frequency
analysis.
