Is there a statistical test for one participant measured many times? Pretty much throughout my undergrad and postgrad, I have always learned statistical models predicated upon things like large subject size. I also know that a lot of repeated measures designs typically work at reducing the error variance of a predictor by removing the error in between time periods. I know that this can sometimes be used as a way to increase the power of a test where there are far less participants in the study.
So to ask what is probably a really stupid question, is there a valid statistical test in which you could use a single participant measured hundreds of times to measure an independent and a dependent variable? I would think that normally this would be an issue in terms of generalizability to the rest of the population, but I'm just curious if there is such a thing in statistics, because I couldn't find something like this when I looked online and I imagine there is good reason for it.

Technically, it depends on the assumptions you are willing to make. You could assume that "the rest of the population" behaves exactly like the one participant that you have, and this then allows generalisation. This, however, is an assumption that will be hard to justify, and by definition you cannot check it using data from other participants (as you don't have them). So normally, in a real situation, this should not be accepted.
What you can use is all the machinery of time series analysis (also depending on certain assumptions that may or may not be justified) that will allow you to predict future observations for the same individual (involving tests for things such as the absence of a trend or seasonal variation). For example https://cran.r-project.org/web/packages/funtimes/vignettes/trendtests.html
