When do iterated conditional expectations converge? Take a probability space $(\Omega,\mathcal{F},\mathbf{P})$ and random variable $X$ satisfying $\mathbf{E}[|X|]<\infty$.
Define the iterated expectations of X as follows: $X_0 = X$, and, inductively, $X_k = \mathbf{E}[X_{k-1} \, | \, \mathcal{G}_k]$, where $\mathcal{G}_k \subseteq \mathcal{F}$ is some $\sigma$-algebra. Assume that all the $\mathcal{G}_k$ come from some finite set of $\sigma$-algebras.
In words, we are repeatedly taking conditional expectations with respect to various information.
Can we conclude that the sequence $(X_k)_k$ converges $\mathbf{P}$-almost surely? 
In my comment on the question I include some of what I know about this, but I suspect that this is easy if one looks at it right, or covered in a standard reference. 
 A: Here's what's known, courtesy of Omer Tamuz.
Here's what's known, with references below.
Amemia and Ando (1965) prove weak convergence in $L^2$. This also covers the finite-$\Omega$ case. Convergence in norm in $L^2$ seems to still be open. It is known to converge if your sequence of algebras is periodic, as is shown by Delyon and Delyon (1999). If you more generally allow projections to a closed subspaces of $L^2$, then there is a recent example by Kopecká, E. & Paszkiewicz (2017) where no convergence in norm occurs. But not every projection to a subspace corresponds to conditional expectation according to some sub-sigma-algebra.
Amemiya and T. Ando, Convergence of random products of contractions in Hilbert space, Acta Sci. Math. (Szeged) 26 (1965), 239–244.
B. Delyon and F. Delyon, Generalization of von Neumann’s spectral sets and integral representation of operators, Bull. Soc. Math. Fr. 127 (1999), 25–41.
Kopecká, E. & Paszkiewicz, A, Strange products of projections, Israel Journal of Mathematics, April 2017, Volume 219, Issue 1, pp 271–286
A: M. Akcoglu and J. King. An example of pointwise non-convergence of iterated conditional expectation operators. Israel J. Math. 94 (1996)
