I want to construct a process $(Z_n)$ adpated to a filtration $\mathcal{F}_n$ such that $$E[Z_{n+1}\mid Z_n]=Z_n, E[Z_{n+1}\mid\mathcal{F}_n]\not=Z_n$$
I start by taking the three values $1,2,3$; $Z_1=X_1, Z_2=X_1+X_2$ and now I want to construct $Z_3-Z_2$ independent of $Z_2$ but determined at times $1$ and $2$. How can I do that?
To continue your construction, take $X_1,X_2$ as independent standard normal distributions, then since $cov(X_1+X_2, X_1-X_2) = 0$ and combined together they are a Gaussian vector, we have $X_1 + X_2$ is independent of $X_1 - X_2$, so let $Z_3 - Z_2 = X_1 - X_2$, which is independent of $Z_2$ but determined at time 1 and 2.
Another way to give an example is to let $Z_{n+1} = Z_n + W_n$, where $W_n$ is a centered r.v. $\mathcal{F}_n$-measurable and independent of $Z_n$, then $E[Z_{n+1}|Z_n] = Z_n +E[W_n|Z_n] = Z_n +E[W_n] = Z_n$ and $E[Z_{n+1}|\mathcal{F}_n] = Z_n + W_n$