A super-martingale $\{X_n\}$ in discrete time is usually represented as having the defining property
$$X_n \geq E[X_{n+1} \mid \mathcal F_n] ,\;\; \forall \,n \tag{1}$$
where $\{\mathcal F_n\}$ forms an increasing sequence of $\sigma$-fields.
As pointed out in a comment to the original form of this question, this is equivalent to
$$X_n \geq E[X_{k} \mid \mathcal F_n] ,\;\; \forall \,n \leq k \tag{2}$$
which is the one providing the discrete analogue for the analogous condition for continuous time.
By the tower property of conditional expectation we have
$$E[X_{n+2} \mid \mathcal F_n] =E\Big( E[X_{n+2} \mid \mathcal F_{n+1}]\mid\mathcal F_n\Big) \leq E[ X_{n+1}\mid\mathcal F_n] \leq X_n \tag{3}$$
Verbally, standing at any period and looking two periods (or in fact any number of periods) ahead, we expect a lower value than the current one.
Consider now the following situation:
Let a (deterministic for simplicity) non-negative sequence $\{a_n\}$ for which we have that some individual elements of the sequence are smaller than unity, but also that
$$\forall n,\; \exists\; s(n) : \forall s \geq s(n),\; \prod_{j=1}^{s}a_{n+j} \geq 1 \tag{4}$$
Verbally, if you multiply enough of them, their product will eventually become equal or greater than unity.
Assume also that the following relation holds:
$$X_n = a_{n+1}E[X_{n+1} \mid \mathcal F_n],\;\; \forall\, n \tag{5}$$
Solving forward, this gives
$$X_n = \left(\prod_{j=1}^{s}a_{n+j}\right)E[X_{n+s} \mid \mathcal F_n] \tag{6}$$
Given the properties of the $\{a_n\}$ sequence, we cannot say that $\{X_n\}$ is a supermartingale, because $(1)$ may not hold $\forall\, n$, or $(2)$ may not hold $\forall n\leq k$. But the following does hold :
$$ \forall n, \forall s \geq s(n), X_n \geq E[X_{n+s} \mid \mathcal F_n] \tag{7}$$
Verbally, standing at any $n$, if we look far enough into the future, we eventually expect to see consistently lower values of the $\{X_n\}$ sequence than the current one. But, standing at any $n$, there is an initial finite future period (for the index running from $n+1$ to $n + s(n) -1$), for which the supermartingale property may not hold.
From my original three questions, this one was the essential one:
Does some kind of convergence obtains for $\{X_n\}$ when $(4)$ and $(5)$ hold?
The process is not a supermartingale, but when a core property is violated only for a finite number of periods, and holds there after, the asymptotic results sometimes still hold. Although in our case, the core property is violated for a finite number of periods always, i.e. $\forall n$.
Note: A simple example of a process like $X_n$, is an $AR(1)$ with non-stochastic, time-varying autocorrelation coefficient whose sequence exhibits property $(4)$.