The dynamic properties of the peroxidase-oxidase reaction are investigated in a continuous flow stirred tank reactor (CSTR) in the free running mode. We report the first experimental period-doubling route to chaos in a Farey sequence by varying the O-2 flow. From a 1(1) starting pattern, period doubling, chaos, and subsequently 1(2), 1(3) and 1(4) oscillations are observed. The chaotic state is clearly identified by the one-dimensional map and the fractal nature of the Renyi dimensions. It also shows an increased dissipation, compared with the dissipation of the periodic time series. A slight change in pH leads to the observation of another Farey sequence containing a concatenated state but no chaos. Model calculations are compared with the experimental results.