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There's one point on the globe that satisfies this condition trivially: the north pole. However, consider the points 1 + 1/(2*pi) miles away from the south pole. Call point A any of these (infinite number of) points. Go a mile south to point B. When you go a mile east, you end up back at point B (you travelled once through every line of longitude). A mile north then brings you back to point A.
There are points still closer to the south pole such that going a mile east brings you through each line of longitude exactly twice, three times, or as many times as you want. Thus we have an infinite number of concentric rings of infinite numbers of points, and we can start a mile north of any of them.