Construction

Let $(W_i)_{i\ge 1}$ be independent nonnegative random variables such that

$$ \mathbb E[W_i]=1 \qquad \text{for all } i. $$

Define

$$ X_0=1, \qquad X_n=\prod_{i=1}^n W_i, \qquad \mathcal F_n=\sigma(W_1,\dots,W_n). $$

Then $(X_n)$ is a martingale with respect to $(\mathcal F_n)$.

Verification

Each $X_n$ is $\mathcal F_n$-measurable. Also,

\prod_{i=1}^n \mathbb E[W_i] =1, $$

so $X_n$ is integrable.

For the martingale property,

$$ X_n=X_{n-1}W_n, $$

hence

X_{n-1}\mathbb E[W_n\mid \mathcal F_{n-1}]. $$

Because $W_n$ is independent of $\mathcal F_{n-1}$,

$$ \mathbb E[W_n\mid \mathcal F_{n-1}]=\mathbb E[W_n]=1. $$

Therefore,

$$ \mathbb E[X_n\mid \mathcal F_{n-1}]=X_{n-1}. $$