This question is cross-posted at MSE with a soon to expire bounty that hasn't generated much discussion.

Let $(\Omega, \mathcal{F},P)$ be a probability space and $(\mathcal{F}_n)_n$ a filtration that increases to $\mathcal{F}$.

Is there a way to quantify the "rate of convergence" of $\mathcal{F}_n \uparrow \mathcal{F}?$

I'll now try to clarify the question by explaining its motivation. I was wondering what can be said about the rate of convergence in Levy's martingale convergence theorem. By that result, for an integrable random variable $X$ we have $$E(X \mid \mathcal{F}_n) \to X$$ almost surely. I was wondering if we can make a statement like $$P(|E(X \mid \mathcal{F}_n) - X| > \epsilon) = O(n^{-a})$$ under fairly general assumptions.

As pointed out to me by Bananach in this question, it's hopeless to expect the rate to be independent of the particular filtration because we can replace $\mathcal{F}_n$ with $$\bar{\mathcal{F}_n} = \mathcal{F}_{\sqrt{n}}$$ (rounding to the nearest integer), and obtain a new filtration for which conditional expectations converge more slowly.

But perhaps if we knew "how quickly" $\mathcal{F}_n$ increases to $\mathcal{F}$, we could find a rate of convergence of conditional expectations that depends on the "rate of convergence" of the filtration.

An example is given in Michael's answer to the same question that I linked to above. Let $X$ be uniformly distributed on $[-1,1]$, assume $\mathcal{F} = \sigma(X)$, and define $$Z_n = X \mathbf{1}_{|X|>2^{-n}} \ \ \text{and} \ \ \mathcal{F}_n = \sigma(Z_n,...,Z_1).$$ Then, $$E(X \mid \mathcal{F}_n) = X \mathbf{1}_{|X|>2^{-n}} \to X,$$ and $$P(|E(X \mid \mathcal{F}_n) - X)| > \epsilon) = 2^{-n}.$$

Another, trivial, example is $\mathcal{F} = \sigma(X)$ and $\mathcal{F}_n = \sigma(X)$ for all $n$, and then $$E(X \mid \mathcal{F}_n) = X.$$ The filtration converges "instantly" and so do the conditional expectations.

The examples suggest that the rate of convergence of the conditional expectations is the same as the "rate of convergence" of the filtration. Can this idea be made precise in general?