Let $X$ be a Polish space. If it allows for an interesting answer, you may assume $X$ is compact or even $X=[0,1]$. The space $\mathcal{P}(X)$ of Borel probability measures on $X$ is also Polish (via the Prokhorov metric). Measures on $\mathcal{P}(X)$ (i.e. elements of $\mathcal{P}(\mathcal{P}(X))$) arise, for example in the ergodic decomposition. I'm looking to understand $\mathcal{P}(\mathcal{P}(X))$ better, especially through examples.
Q1. Is there a natural example of an element $\mathcal{P}(\mathcal{P}(X))$? Q.1.5 What about when $X=[0,1]$?
Q2. What results in mathematics use or refer to an element of $\mathcal{P}(\mathcal{P}(X))$? I'm aware of the ergodic decomposition and its special case, de Finetti's theorem.
Q3. Where is $\mathcal{P}(\mathcal{P}(X))$ studied? I'm aware of Billingsley's Convergence of Probability Measures and Parthasarathy's Probability Measures on Metric Spaces.
EDIT Regarding Q2, I'm most interested in classical results. The answers given by @NateEldredge and @MichaelGreinecker, while interesting and helpful, seem to regard more modern (i.e. not classical) results. I realize that 'classical' is vague, and I'll try to make what I'm after more precise if necessary. The ergodic decomposition is something I consider 'classical'.