I have a random number generator that generates a random real number between $0$ and $1$ (not including $0$ or $1$).
I generate a first number. Then I generate more numbers until I get a number that is smaller than the first number.
Let $X=$ the number of numbers generated after the first number. So the possible values of $X$ are $1, 2, 3, ...$.
We can show that $E(X)=\infty$.
$P(X=n)$
$=P(\text{smallest number is last, and second smallest number is first})$
$=\left(\frac{1}{n+1}\right)\left(\frac{1}{n}\right)$
$\therefore E(X)=\sum\limits_{n=1}^\infty n(\frac{1}{n+1})(\frac{1}{n})=\frac12+\frac13+\frac14+\cdots=\infty$
How can the expectation of $X$ be larger than every possible value of $X$?