Let $(A_{n})$ with $n\geq 1$ an non decreasing sequence of random variable such that $A_{n+1}$ dépend of $X_1,...,X_n$ for all $n$. I agree that $$\mathbb E[A_{n+1}\mid X_1,...,X_n]=A_{n+1}$$ But I don't understand why $$\mathbb E[A_n\mid X_1,...,X_{n}]=A_n.$$
Is it a consequence of the fact that $A_n$ doesn't depend of $X_{n}$ and thus
$$\mathbb E[A_n\mid X_1,...,X_{n}]=\mathbb E[A_n\mid X_1,...,X_{n-1}]=A_n\ \ \ ?$$
$A_n$ depends on $X_1, ..., X_{n-1}$, therefore it is $\sigma(X_1, ..., X_n)$-measurable, because $\sigma(X_1, ..., X_{n-1}) \subset \sigma(X_1, ..., X_n)$. Also, there is no need for $A_n$ to be nondecreasing.