I know a few stochastic processes that are described as 'distributions over distributions'. For example, the Dirichlet process draws discrete distributions from a default model (see e.g., p3 of these notes) .
The discrete distributions could be made continuous by assuming some kinds of kernels. E.g., if you have a discrete distribution with a probability $p_i$ associated to a variable $x_i$, you could make a continuous distribution from Gaussian kernels, $$ p(x) = \sum_i p_i \cdot g(x | \mu = x_i, \sigma^2 = 1) $$ where $g$ is a Gaussian.
Furthermore, although the discrete distributions from a Dirichlet process are a 'small' set of the continuous ones, it can be argued that they are a 'large' set in the weak or pointwise sense (see e.g., p4 of these notes). Thus, a Dirichlet process could be used to relax a parametric model in a non-parametric way.
However, are there any distributions over continuous distributions? That is, is there e.g. a stochastic process like the Dirichlet one that covers a set of continuous distributions in a strong rather than weak sense? Is there e.g. a meaningful continuum limit of the Dirichlet process that could be used for this purpose? Are there other possibilities?