Proposition: Let $\kappa$ be some infinite number cardinal number. There exists a probability space $(\Omega,\Sigma,\nu)$ that carries $\kappa$ independent random variables with uniform distribution on $[0,1]$ and such that such for every family $\langle g_i\rangle_{i\in I}$ of real-valued random variables with $\#I\leq\kappa$ and every probability measure $\mu$ on $\mathbb{R}^I\times\mathbb{R}^J$ with $\#J\leq\omega$ and $\mathbb{R}^I$-marginal equal to the joint distribution of $\langle g_i\rangle_{i\in I}$, there exists a family of random variables $\langle g_i\rangle_{i\in I}$$\langle g_i\rangle_{i\in J}$ such that the joint distribution of $\langle g_i\rangle_{i\in I\cup J}$ equals $\mu$. $$~$$
One can take $\Omega={0,1}^{\kappa^+}$, $\Sigma$ the product-$\sigma$-algebra, and $\nu$ the fair coin-flipping measure. The proposition can be proven using ideas from this paper.
The proposition shows that one can find a probability space that can carry a lot of nontrivial random variables and such that one can always add ex-post a countable number of random variables at a time whose distribution relates in any way to the other random variables. One never runs out of space; there is no need to enlarge the underlying probability space.
This is probably more than enough for any reasonable probabilistic argument, but works with only set-many random variables. If one wants to do this with random variables indexed by the class of ordinals, one could do this by viewing the class of all sets as a genuine set in a larger universe that contains a strongly inaccessible cardinal; this seems to be the preferred method of foundation-conscious category theorists for dealing with similar size problems.
