When reading about the central limit theorem, the concept of infinite population mean arises.How can a population mean be infinite?
Two examples of random variables $Y$ with 'infinite' $E(Y).$
1) Let $X$ be geometric with $P(X = i) = 1/2^i,$ for $ i = 1, 2, \dots.$ Then let $Y = 2^X.$ If you try to find $E(Y)$ you get an infinite sum with all terms equal to 1.
2) Let $X \sim T(1),$ Student's t distribution with one degree of freedom. Then let $Y = |X|.$
Very roughly speaking, the tails need to be 'fat' enough that $\sum |x|f(x)$ or $\int |x|f(x)\,dx$ (taken over all values) is infinite, but not so fat as to prevent $\sum f(x) = 1$ or $\int f(x)\,dx = 1$ (or you wouldn't have a legitimate distribution).
Addendum: Also, relevant to conditions for the CLT: It is possible to have a random variable $Y$ for which $E(Y)$ exists and $Var(Y)$ does not; $T(2)$ is one example.