  Feigenbaum universality is shown to occur in subcritical shear flows. Our
testing ground is the counter-rotation regime of the Taylor-Couette flow, where
numerical calculations are performed within a small periodic domain. The
accurate computation of up to the seventh period doubling bifurcation, assisted
by a purposely defined Poincar\'e section, has enabled the estimation of the
two Feigenbaum universal constants with unprecedented accuracy in a fluid flow
problem. We have further devised a method to predict the bifurcation diagram up
to the accumulation point of the cascade based on the detailed inspection of
just the first few period doubling bifurcations. Remarkably, the method is
applicable beyond the accumulation point, with predictions remaining valid, in
a statistical sense, for the chaotic dynamics that follows.
