Suppose $Y_1$ and $Y_2$ are i.i.d. random variables with a lognormal distribution. That is, for $i \in \{1, 2\}$
$$ Y_i = \exp(X_i) $$
where each $X_i$ is a normal distribution with mean $\mu$ and variance $\sigma^2$. Then, by independence:
$$ E[Y_1 Y_2] = E[Y_1]E[Y_2] = \left( \exp\left(\mu + \frac{1}{2}\sigma^2\right)\right)\left( \exp\left(\mu + \frac{1}{2}\sigma^2\right)\right) = \exp\left(2 \mu + \sigma^2\right)$$
At the same time:
$$ E[Y_1 Y_1] = E[Y_1^2] = \exp\left(2 \mu + \frac{1}{2} 4 \sigma^2\right) = \exp\left(2 \mu + 2 \sigma^2\right) \neq E[Y_1 Y_2] $$
When I first realized that $E[Y_1 Y_1] \neq E[Y_1 Y_2]$, I thought I had stumbled on a paradox or an impossibility. I'm guessing the reason these values aren't the same is that $Y_1$ is not independent of itself, even if is identically distributed to itself, but it is independent and identically distributed to $Y_2$.
Is that all that's going on here? Does anyone have any deeper intuition about what at first glance looked like a paradox to me?