Regions, Secants and Edges

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In a circle, I have N points then I would make secants and edges then I have regions, which has sides as edges. My questions is when N increases, the regions double but it will not last long, so why regions cannot continue to double?

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I will assume that the reason why you think the regions should double is similar to something like "every old region should be divided into two regions when a new point is added."

However this is not true. It does not take much effort to realize that all the regions that are originally staying on the circumference of the circle except the one where the new point is added will not be splitted into two regions.

This also means there must be regions that are splitted into three or more regions as well so that it can "look like" doubling for small N.

As N grows big, the number of regions that are splitted into three or more and the number of regions that are not splitted at all do not balance out each other any more so the total number will not double itself any more.

What really matters is the number of edges intersected when a new point is added as shown in the wikipedia page https://en.wikipedia.org/wiki/Dividing_a_circle_into_areas.