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.

There are statistical methods for hierarchical or longitudinal data, where for example you have a multiple measures on each individual.  These methods recognize that multiple measurements on an individual can't be treated the same way as single measurements on many individuals because they aren't independent observations.  If you had 100 blood pressure measurements on 2 subjects, treating them as independent observations with ordinal least squares would produce artificially low estimates of standard error.  Despite 200 data points, you actually know very little about the overall variance of blood pressure over the population.
The methods are called mixed models, hierarchical models, or random effects models.
https://online.stat.psu.edu/stat510/lesson/10
