Fair game $\neq$ constant expectation
One always thinks of a martingale as a "fair game". However, this is not the same concept as a process that has constant expectation. For example, let $(S_n)_{n \geq 0}$ be the symmetric random walk on the integers. Then the process $$X_n := S_n^3$$ has constant expectation (equalling zero for all $n$), but it is not fair in the following sense. If a casino ran a game where you win $X_n$ when you stop at time $n$ (but you don't have to stop deterministically), there is a strategy where you can win money in finite time.
Specifically, let $\tau = \inf \{n \geq 0: X_n \in \{-1,8\}\} \wedge 2$. Then
\begin{align*} \mathbb{E}X_{\tau} &= \mathbb{P}(\text{hit -1}) (-1) + \mathbb{P}(\text{hit } 8) (8) + \mathbb{P}(\text{hit neither}) (0) \\ &= \frac{1}{2}(-1) +\frac{1}{4}(8) +\frac{1}{4}(0) \\ &= \frac{3}{2}. \end{align*}
In other words, leave the casino if the value is $-1$ or $8$; but if neither value is hit by time 2 (which happens iff $X_1=1,X_2=0$), then just leave. In this way your expected winning is $3/2$.
The existence of a winning strategy is guaranteed by the optional stopping theorem, which says that an integrable process $(X_n)$ is a martingale if and only if for all bounded stopping time $\tau$ such that $\mathbb{E}X_\tau = \mathbb{E} X_0$. The converse of this is if $X$ is not a martingale, there exists a bounded stopping time $\tau$ such that $\mathbb{E} X_\tau \neq \mathbb{E}X_0$.
Question: Given an integrable process $X$, what methods can one use to construct a bounded stopping time such that $\mathbb{E} X_\tau \neq \mathbb{E} X_0$?