With the use of real-variable techniques, we construct a weight function ω on the interval $[0,2\mathrm{\pi })$ which is doubling and satisfies $log\mathit{\omega }$ is a BMO function, but which is not a Muckenhoupt weight ($A_{\mathrm{\infty }}$). Applications to the BMO–Teichmüller space and the space of chord-arc curves are considered.