There are a number of metrics that are commonly used to settle on a number of factors to extract. There's some judgment in it to be sure, but you are looking at data-driven phenomena as well like percentage of variance explained, a precipitous drop in new added value, and etc.

Essentially you come to a point where extracting another value isn't explaining much more variance or producing a factor that's meaningful (e.g., two items of conscientiousness with negative wording are the entire new "factor" and only take variance explained from 80% to 82%).

https://www.theanalysisfactor.com/factor-analysis-how-many-f...

Like pretty much every unsupervised learning problem, there is no data which can tell you what N should be. If there was, you could trivially use that measure to convert it to a supervised learning problem.

More generally, all of those metrics are pretty fluffy (the scree plot is probably the best, as it doesn't give you a p-value).

In order to actually do this right, you need multiple studies and CFA, but the standard in the field is to refit your exploratory model as a CFA on the original dataset, which is almost certainly causing overfitting in most cases.

Other models actually did settle for higher numbers of factors, but higher numbers were usually found to have categories that could be reasonably collapsed into one of the Big 5.

Fwiw, my understanding is that most psychologists believe that with further study, these categories will map to actual functioning of the brain and nerve systems. For example, "Neuroticism" will somehow map to the excitability of certain nerve systems and "Agreeableness" will somehow map to functioning of our mirror and mentalizing systems. "Conscientiousness" will map to inhibiting functions, "Openness" will map to approach/flee type emotional functions, and "Extraversion" will map to systems like dopamine release on securing rewards for behavior.

That's a belief that's a long way from being testable though.