A distribution whose variance is infinite

Give an example of a probability distribution that does not have a finite variance.

For example, the density $f(x)=2x^{-3}$ on $[1,\infty)$. Its mean is $\int_1^\infty 2x^{-2}dx=2$, but $\mathbb{E}[X^2]=\int_1^\infty 2x^{-1}dx$ diverges. A Student $t$ with $2$ degrees of freedom also works; the Cauchy distribution is a further example, though its mean is undefined as well.