Generalization of probability

Suppose I am not only interested in a certain probability of an event $p \in [0,1]$, but have a "deeper uncertainty", so to speak, which I would model as a CDF of probabilities $F(p) \in [0,1] \to [0,1]$. Does this extend indefinitely to $F(F(p)) \in ([0,1] \to [0,1]) \to [0,1]$, and so on? How can I deal with the infinite dimensions of the spaces of functions and spaces of sets of functions, etc?

For example, imagine I want to bet on the outcome $X$ of a coin, which is $X=0$ if it comes out heads and $X=1$ if it comes out tails. My bet would not be effected if I restrict myself to the probability of heads, which can be obtained by integrating out $f_P(p)$ by taking $$p^*=\Pr(X=1)=\operatorname{E}[P]=\int_0^1 p f_P(p) \, dp.$$

Now if I instead toss the same coin multiple times, I can update my estimate of $p$ using $F(p)$, which may be obtained by integrating over $F(F(p))$ -- but what does this space (of all possible $F(p)$) even look like?

Edit: To further clarify, suppose we assign a density $f_{F_P}(F_P)$ to each CDF. We can obtain a cdf of probabilities by integrating out $f_{F_p}(F_P)$

$${f}^*_p(p)=\lim_{\delta \to 0} \frac{1}{\delta} \Pr(p \le P \le p+\delta)=\int f_P(p)f_{F_p}(F_p)dF_P$$

But the space spanning all possible $F_P$ has infinite dimension. How to proceed? (for example, what would an uniform distribution over $F(F_p)$ look like?)