Is it possible to have too many priors in Bayes Theorem? Each prior restricts the sample space, so in theory, your probability becomes more and more accurate. However, if our sample space becomes too small, the tradeoff between the accurate sample and the confidence we gain from having more data points becomes apparent.
Example: Assume I want to know the odds that I will contract COVID. I ask myself what is $P(\text{getting covid} | \text{20 years old})$. Let's say I get $10\%$. Then I stack another prior to get even more accurate data. I ask myself what is $P(\text{getting covid} | \text{20 years old} \land \text{is not obese})$. Great. Even more accurate probability.
However, what if I ask myself $P(\text{getting COVID} | \text{I have a tattoo of my name on my arm})$. Let's say there are $2$ people in the world, me and my friend, who have a tattoo of my name on his/her arm. Let's say my friend got COVID. This gives me a $1$ in $2$ odds of getting COVID which dosen't make sense.
A couple of ideas - maybe Bayes Theorem should be only used as a tool, maybe the priors have to be relevant to the outcome, maybe there's a threshold below which our sample is not significant. Not sure.

The first thing to realise here is that "probability" does not have the same meaning for everybody. P(getting Covid|20 years old)=0.1 may mean that you personally, based on all you know, would pay up to 10 Euros for the chance of winning 100 Euros in case a randomly selected 20 years old person catches Covid (obviously assuming that all is well defined, so for example this may mean "in the next year" or "before the 21st birthday" or whatever; same is assumed for the other possibilities below). This can be referred to as subjective epistemic probability.
It may also mean that in the long run, observing all 20 years olds, the relative frequency of those who catch Covid will approach 10%. Note that this is somewhat idealised, because we can only ever observe finitely many people. That would be a frequentist probability.
There is also objective epistemic probability, which wouldn't refer to individual persons but rather involve only secured knowledge and otherwise "informationless" priors, and which are not motivated via betting rates but rather via axioms for rational reasoning.
Bayes theorem holds for all those versions of probabilities (even though some use terminology so that it implies that "Bayesian probabilities" are distinctively different from frequentist ones), but in a given situation you need to be clear about which one you mean.
Talking about P(getting COVID|I have a tattoo of my name on my arm), in a situation where you know that currently there are only two people and one of them has caught Covid whereas the other one has reached the 21st year without catching it (assuming this is how you define the event), this will not tell you the probability, but rather the relative frequency of that (conditional) event. Assume that the probability refers to a potential future situation in which somebody else will have that tatoo (and no further information given).
Thinking about subjective probability, you are still free to choose your probability according to your beliefs (as Dave Harris has explained), take into account all your need and also the relative frequency from the sample you know, but 2 is of course not a big sample size, so you may not give that result a large influence on your subjective probability choice. For sure you can play the betting game, and for sure you are not constrained to choose your betting rate as 0.5.
Thinking about frequentist probability, the probability refers to the hypothetical event that infinitely more persons occurs with your tatoo. The relative frequency $\frac{1}{2}$ then is an estimate of the assumed true probability, but every good frequentist would tell you that it can't be a very good estimate with a sample size of two (as can be reflected in wide confidence intervals). Furthermore, if in fact you believe that in reality there will be no further people with that same tatoo, the frequentist probability can be seen as a useless fiction, because it refers to the repetition of an "experiment" that in reality will never be repeated.
Note however that we can think of indirect repetition in the sense that we may have a regression model explaining some $y$ from many $x$-variables; then a specific vector of values of all the $x$-variables may never repeat precisely, but we may think of observations with other $x$-values as repetitions of the same process, the same mechanism how the $y$ depends on the $x$; formally, probability refers to the "error term" implied in the model, which  is repeated even if the precise $x$ is not.
Regarding the objectivist epistemic probability, given that there is no other information, you could start from an informationless prior about the probability and then update it based on your observed sample of size 2. The resulting posterior can be used as the prior for the next observation. This seems pretty nice; trouble is that people in the literature have come up with more than one way of specifying the "informationless prior"; however you choose it, it will have some implications that some can be seen as representing some kind of "information", which we'd want to avoid. Also if there is in fact more information that does not come in a nice formal form, it is often not clear how to "objectively" involve it.
What you are right about is that the more precisely you specify the conditioning event, the smaller is the number of available cases (even potentially in the future as long as we think about real observations rather than an idealised fantasy world). This is in fact something of a dilemma for all probability interpretations, unless you feel confident to just make up a subjective value out of thin air, because all further considerations require assumptions (regression models, independence, absence of interactions etc.) that cannot be checked at any precision using available observations for the exact event you are interested in.
