  We demonstrate that "natural inflation", also known as "axion inflation", can
be compatible with Planck 2018 measurements of the cosmic microwave background,
while predicting an exponentially small tensor-to-scalar ratio, e.g., $r\sim
10^{-15}$. The strong suppression of $r$ arises from dynamics of the radial
component of the complex scalar field, whose phase is the axion. Such tiny
values of $r$ remain well below the threshold for detection by CMB-S4 or Simons
Observatory B-mode searches. The model is testable with the running $\alpha_s$
of the spectral index, which is within reach of next-generation CMB and
large-scale structure experiments, motivating the running as a primary science
goal for future experiments.
