Plausibly? The 12th-century Russian type of land dividing (a ruler divides his land among his children, who divide it among their children, etc etc until you have hundreds of small states) is common in other parts of Europe (just like it was in France a few centuries before, and I think the HRE also splintered in a similar way), and the Mongols never come to crush it. Come 1400, you have thousands upon thousands of tiny states all across Europe. That's probably when the situation about stabilizes - in a way reminiscent of Puss in Boots: not much left to divide between children.
Then, such a situation leads to very minor colonising (not much money and especially materials). Indians (both American and Asian), Africans and Oceanians survive in their small tribes.
Of course, there would inevitably be a guy who would conquer a lot of land... comparatively. (Like how the whole Athens vs. Sparta thing, while a lot to them, is barely noticeable on a map of Europe.) And his children, and their children in turn, would divide even that. Balance, you know.
The only blemish on the whole thing might be the existence of China... which, of course, still is 1) isolationist and 2) often divided among warlords anyway. So that also doesn't matter.
Oh, and I don't know what would happen to such a world when the Industrial Revolution comes... if it does. Maybe it doesn't. I'm not very good at predicting such things.
How many divisions there are? Don't know. Ten thousand as a wild guess, most of them in Europe.
Less plausibly: Culvana from Thande's Moonstruck.
More plausibly and/or with a later POD: not much more than a few hundred. Space-filling countries are inevitable in some places (e.g. Kiribati might not have its OTL borders, but what there certainly won't be is each little archipelago as its own country; North Siberia and Nunavut, and for that matter Greenland, would be either affixed to whoever wants to own them, or at most divided between two or three such countries; that's certainly not the only cases). Still, a surviving HRE might well double that number, and a princely-state India (or a fragmented USA/Brazil/Mexico/whatever) add another bunch; the end result is probably close to a thousand. Maybe a little more - but not by much.
...So what, how?

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