Update: Here is an update to my answer based on the details that you gave about your project in your comment below. It looks like you would like to consider a space $X$, and then the space $\mathcal{M}_1(X)$ of probability measures on $X$, and then the space $\mathcal{M}_1(\mathcal{M}_1(X))$ and so on.
This is doable from a computational prospective assuming $X$ is a computable metric space and $\mathcal{M}_1(X)$ is the space of Borel probability measures on $X$. Examples of computable metric spaces include $\{0,1\}^\mathbb{N}$ (good for coin flipping), $C([0,1])$ (good for Brownian motion and continuous path processes), the RCCL functions on $[0,1]$ under the Stockenrod metic (good for RCLL processes), $R^\mathbb{N}$ (for real-valued discrete time Markov chains) and much more.
Given a computable metric space $X$, the space $\mathcal{M}_1(X)$ of Borel probability measures is also a computable metric space. The metric is the Levy-Prokohorov metric (which is a bit of a pain to work with). Also, one can represent the measures by their integrals, which are continuous linear operators on the bounded continuous functions. This represents the same topology. A good resource is "Computability of probability measures and Martin-Löf randomness over metric spaces" by Hoyrup and Rojas (you can skip the stuff about randomness for your needs) and "Representing Probability Measures using Probabilistic Processes" by Matthias Schröder and Alex Simpson (mentioned in another answer). The too papers have equivalent approaches.
Indeed, while I don't know if epistemic game theory has been looked at from a computational prospective, I know that related concepts like conditional probability, de Finetti's theorem, and nonparametric Bayesian statistics have been looked at using the theory of computable metric spaces and computable probability measures. Dan Roy's website is a good place to begin.
