Let $X_t$ be a positive discrete time martingale. (I.e., $X_t>0$ and $\mathbb{E}_t X_{t+1}=X_t$ for all $t\in\mathbb{Z}$.)
Then by Doob's supermartingale convergence theorem, there exists some non-negative random variable $X_\infty$ with $\mathbb{E}X_\infty<\infty$ such that $X_t$ converges almost surely to $X_\infty$ as $t\rightarrow\infty$.
It is easy to come up with examples in which $X_\infty=0$ (for example, $X_t=X_{t-1} \exp(\sigma \varepsilon_t - \frac{\sigma^2}{2})$, where $\varepsilon_t\sim\mathrm{N}(0,1)$).
Can anyone give a simple analytic example in which $0<\mathbb{E} X_\infty^2-(\mathbb{E} X_\infty)^2<\infty$? [Now answered.]
Follow on question: Can anyone give a simple analytic example in which $0<\mathbb{E} X_\infty^2-(\mathbb{E} X_\infty)^2<\infty$ and in which $\frac{X_t}{X_{t-1}}$ is a stationary process?
Let $\mathbb{P}(Z = 2) = \mathbb{P}(Z=0) = \frac 12$, $X_0 = 1$, and $X_t = Z$ for $t > 0$. $X$ is a martingale with respect to its own filtration, and $\lim_{t \rightarrow \infty} X_t = Z$ satisfies $\mathbb{E}[Z^2] - \mathbb{E}[Z]^2 \in (0,\infty)$.