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There is no greatest number for a circle.
A circle can have 0, 1, or 2 x-intercepts and 0,1, or 2 y-intercepts, bringing the total to 0, 1, 2, 3, or 4 intercepts.
A straight line in Cartesian space will have at most two.
There is no fixed limit. A periodic function, such as the sine function, can have an infinite number of x-inercepts.