I always thought it had both one side and infinite sides.
If it has one side, it is not a polygon, if it has infinite, it is.
When we look at the definition of a circle (a closed plane curve, every point of which is equidistant from a fixed point within the curve) we can see that it does indeed have points which are connected.
However, if it had infinite sides, all of the connective lines between points would have to be of an infinitely small size in order to fulfill it's equidistantness.
Does a line exist if it has no dimensions? Theoretically, maybe, but in our reality, it does not.
So I guess that realistically, a circle must be consistent of one rounded side, but perhaps theoretically, there could be an argument.