The actual formula is O(n^6) scaling, and for circles it's very easy to achieve that. So a circle divided into 16ths has 4096 times less error than one in quarters and one made of 128 nodes is a (binary) billion times better.

This is true for curve fitting of arbitrary curves, not just circles, though arbitrary curve fitting is a bit tricky (it requires solving quartic equations). I personally think you can take this as evidence that cubic Béziers are "good enough" for all practical tasks requiring curve representation, though for specific applications there might be other representations that work better.