Suppose $X_t$ is a submartingale, with the property that it has "large enough" increments: $$\mathbb{E}[X_{t+1}-X_t|X_t] \geq \exp(-X_t^2/t),$$ and it is not uniformly bounded, but each $X_t$ is bounded.
For a given threshold $\theta$ define the stopping time $\tau = \inf\{t: X_t \geq \theta\}$. The goal is to lower bound the probability $\Pr\{\tau \leq T\}$.
The intuition behind this question is that if the submartingale increases enough, it will pass "small enough" thresholds with high probability and in small time.