  We consider the steepest rate at which the power spectrum from single field
inflation can grow, with the aim of providing a simple explanation for the
$k^4$ growth found recently. With this explanation in hand we show that a
slightly steeper $k^5 (\log k )^2$ growth is in fact possible. Moreover, we
argue that the power spectrum after a steep growth cannot immediately decay,
but must remain large for the $k$ modes which exit during a $\sim2$ e-fold
period. We also briefly consider how a strong growth can affect the spectral
index of longer wavelengths preceding the growth, and show that even the
conversion of isocurvature modes likely cannot lead to a stronger growth. These
results have implications for the formation of primordial black holes, and
other phenomena which require a large amplitude of power spectrum at short
scales.
