There is a mathematical point I want to make that I didn't see in the prior "nit-picking". People have mentioned that some of the children will not live to reproduce. The "zeroing" of a population can actually occur after several generations, and I think this is more common than many would expect.
In mathematical terms, you have given in your post an "exponential model" for the growth of a population, say, the descendants of John Smith born 1650. The descendants of John Smith are a value that changes over time, and you could "model" it with all sorts of models. The exponential model is wonderful because a mechanism explains how it arises (i.e., your explanation of how to arrive at 4^13 after 13 generations) and it is very simple, so easy to work with. But any work like this is creating only a model of the world, not describing the world for certain. It is a mathematical abstraction of what will happen, and while the real world is certain to differ from what your model says, you try to pick a model that will most closely describe what happens in the real world. People have already explained how endogamy makes this model unrealistic after enough generations.
One of the most common ingredients to include that complicates the exponential model but also makes it more realistic is an "extinction threshold". This is a threshold where if the population is low enough, and things go wrong, the population goes to zero. There would be some work to implement this (you'd need to move to a stochastic model, blah, blah, ... don't worry about that), but the point is, the long-term behavior of a population will end up doing two drastically different things: growing or heading to 0 and then staying put. It may take many generations to reach 0, but once it gets there, it is certain never, ever to grow.
My point is that the process of going to 0 can take more generations than one would expect. I offer as an example Susan Jackson
Susan was probably born around 1800. She was one of six illegitimate children of James Jackson of Ballybay, a scion of a fabulously wealthy family from County Monaghan in Ireland in the 1700s. They were politically active, and most of the family fled in the late 1790s and early 1800s.
I am descended from Nothern Irish Jacksons, and a fourth cousin on my line found a letter written to her aunt from a woman named Lillian Virginia Koller in the early 1900s and sent it to me. It was clear that it contained a transcription of the family bible of James Jackson of Ballybay, so I was very excited. No one seems to know this bible existed, or at least existed within memory.
With some help, I eventually traced Lillian's ancestry back to Susan. Then I went looking for living descendants of Susan hoping to find one who may know about the bible, and so far as I can tell, there may be none, with the last descendants passing away around 1950. You can see that she had quite a few descendants:
(Actually, I didn't create profiles for them all because it got got depressing creating so many profiles for terminal lines.)
I have written to the Staten Island Historical Society Museum to see if the bible ended up in their collections. They said that it is common for the last known scion of a house to pass away with a bible and that if no other relatives can be found, it may be bought at auction by a museum or historical society.
P.S. I have done a lot of work on identifying people in the 1774 Rhode Island Census, which quite often includes going back and looking for ancestors into the 1600s. There are many such New England profiles that do not exist yet on Wikitree.