$L^1$ version
If $(X_n)$ is a martingale and $a>0$, then
$$ \mathbb P\left(\max_{0\le t\le n}|X_t|\ge a\right) \le \frac{\mathbb E[|X_n|]}{a}. $$This controls the whole path maximum using only the terminal value.
$L^p$ version
If $p>1$, then
$$ \mathbb E\left[\max_{0\le t\le n}|X_t|^p\right] \le \left(\frac{p}{p-1}\right)^p\mathbb E[|X_n|^p]. $$Common consequence
If
$$ \sup_n \mathbb E[|X_n|^2]<\infty, $$then
$$ \mathbb E\left[\sup_{n\ge 0}|X_n|^2\right]<\infty. $$