1 BU ×1 BU plane, 10 levels of Subdivision Surface modifier set to Catmull-Clark:
That's not a circle.
What is it?
(Mathematically, what's the limit surface of a Catmull-Clark subdivision?)
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For rectangular control-point meshes the Catmull Clark subdivision modifier creates a standard B-spline surface. Regardless of the number of subdivision it will not converge to an accurate circle, but with more control points the difference between a circle and the interpolated shape will be less noticeable.
The limit of a Catmull-Clark surface is the union of (infinitely many) uniform cubic B-spline patches. The "infinitely many" bit is in parentheses because that's only true when the surface contains singularities (that come from non-4-valent, "extraordinary" vertices).
For the curve case, it's much simpler: you just get the uniform cubic B-spline that is defined by the control points you started with. Spline means "piecewise polynomial", so this means the limit is a collection of parametric polynomials, joined together (with continuity of curvature) at positions called knots.
One way of seeing that this is true is that each subdivision step is actually performing knot insertion on the original curve, splitting each cubic piece into two and (roughly) doubling the number of knots in the spline.