Are there any random variables so that $\mathrm{E}[XY]$ exists, but $\mathrm{E}[X]$ or $\mathrm{E}[Y]$ doesn't? Are there any random variables so that $\mathrm{E}[XY]$ exists, but $\mathrm{E}[X]$ or $\mathrm{E}[Y]$ doesn't?

Sure, let Y be the value $2^n$ with probability $\frac{1}{2^n}$.  Let $X=1/Y$.  Then $XY=1$ and $E(X)=\frac{4}{3}$ exists, but $E(Y)$ does not exist.
