Ring oscillator frequency
That's because Barkhausen only applies to linear circuits. Ring oscillators are made from inverter stages that have so much gain that they have to be treated as nonlinear (saturating) elements.

Two inverters (or any even number of inverters) form a bistable system.

A single inverter generally does not have enough delay to oscillate; instead, it drives itself to an intermediate analog state.

Larger odd numbers of inverters have enough delay to oscillate, since the output of each gate reaches saturation before its input changes again.