Example of a Non-negative Martingale Satisfying Certain Conditions

Modifying your example a little, by letting $Y_0$ be any non-negative random variable independent of $\{Y_n\}_{n\ge 1}$ with $E(Y_0)=1$, $$ M_0=\frac12Y_0+\frac 12,\ \ M_n=\frac12\left(Y_0+\prod_{i=1}^nY_i\right),\ \ n\ge 1 $$ is a martingale converging to $\frac 12 Y_0$.

Hint

Convex combinations of martingales are martingale. Your example converges to $0$. modify if to get an example converging to something else, mix the two.

Answer

Let $X_n$ be the martingale you described; $X_n=\prod_{i=1}^n Y_i$, where $Y_i\stackrel{\text{iid}}\sim \operatorname{Unif}\{1/2,3/2\}$.

Let $(X_n')_{n\ge 1}$ be an independent copy of $(X_n)_{n\ge 1}$.

Finally, let $\xi$ be Bernoulli$(1/3)$, independent of the previous variables. Then $$ X_n\cdot{\bf 1} ({\xi=1})+(2-X_n')\cdot {\bf 1}(\xi=0) $$ is a martingale whose expectation is always one, and in the limit equals $0$ with probability $1/3$ and equals $2$ with probability $2/3$.