Economic examples of (sub)martingales
Applying the Law of Iterated Expectations on the defining property of a sub-martingale $E(X_{t+1}|X_1,\dots,X_t）\ge X_t$ we have that

$$E\Big[E(X_{t+1}|X_1,\dots,X_t)\Big]=E(X_{t+1}) \geq E(X_t)$$

So a sub-martingale has also its unconditional expected value non-decreasing. Therefore any economic variable that appears to consistently increase in value is a candidate for being a sub-martingale, since the defining property here conditions only on the variable's past, not on any other factors that may influence it. National Output/GDP could be reasonably modeled as a sub-martingale, since it historically "always" increases, bar catastrophic events.

In general, any economic variable that can be modeled as a random walk with non-negative drift, 

$$X_{t+1} = \alpha + X_{t} + u_t,\;\; \alpha \geq 0$$

is a sub-martingale.