It seems Yvian has accepted the challenge and made a steel man attempt.

Consider the great circle passing through your current location and the Earth's poles, together with the great circle perpendicular to it at the poles. These form 4 lines of longitude, each one of which is experiencing a different day simultaneously (for instance, when it is midnight on your line of longitude, it is 6am, noon, and 6pm on the others). Of course, you might wonder why I would single out these 4 lines of longitude instead of just the one at your current location, giving the traditional 1 day per 24 hours, or all of them, giving infinity days per 24 hours. Of course, it would be ridiculous to say that there is a different day going on in one location and another some infinitesimal distance away from it, so the latter is a non-starter. And the standard 1-day answer ignores the fact that different longitudes do not experience the same day. Counting 4 days occurring at the same time makes sense because then the days are separated by 90-degree rotations of the Earth, and correspond to quadrants of a circle. 90 degrees is the most fundamental angle in geometry, and should be considered the primary unit of rotation, as explained in this video (relevant discussion starting at 4:28).