Generalized Binomial Models

With the standard Binomial probability distribution we consider n trials each with a probability of success p. This can be somewhat generalized to the Beta-Binomial Distribution which is effectively the same, but this time the probability of success is a random variable with a Beta Distribution. My question is:

Could I in theory use a p which samples from some arbitrary probability distribution (continuous or discrete) so long as the support is on [0,1], or is there something special about the Beta distribution which allows for such a distribution?

The Beta is just conjugate to the Binomial which makes life easier in a number of ways. If we are in a Bayesian setting with a Binomial likelihood, there is nothing limiting you, theoretically, from applying any prior distribution on $p$ that you like. The problem is that you may end up with an improper posterior, that is, a posterior distribution that does not integrate to one. So if you choose to use such a prior you should check that your posterior is a proper density. But perhaps most importantly, you should be prepared to defend your use of such a prior in that it should make sense for your specific application.