As in St. Petersburg paradox, let $X:\mathbb Z_{>0}\to\mathbb R$ be a discrete random variable with $\operatorname{Pr}[X = 2^k] = \frac{1}{2^k}$ for all $k\ge 0$ (and $\operatorname{Pr}[X = n] = 0$ if $n$ is not a power of two). Then $\mathbb E[X] = +\infty$ or is undefined, depending on the definition.
Closely related, let $Y:\mathbb Z_{>0}\to\mathbb R$ be a discrete random variable with $\operatorname{Pr}[X = (-1)^k 2^k] = \frac{1}{2^k}$ for all $k\ge 0$ (and $\operatorname{Pr}[X = n] = 0$ if $n$ is not a power of two). Then $\mathbb E[Y]$ is undefined.
What does it mean for repeated experiments?
Does $\mathbb E[X]=+\infty$ implies that one should expect that the more one repeats the experiment, the larger the mean of the observed values is? Of course, defining $N$ copies $X_1$, ..., $X_N$ of $X$, one has $\mathbb E[\frac{1}{N}(X_1+\dotsb+X_N)] = +\infty$ so it seems to have no sense to say that the observed values grow in expectation...
What does $\mathbb E[Y]$ being undefined implies? The observed values seems distributed around $0$: Can we quantify this or at least make some sense of it?
The general question is: The expectation of these variables is either infinite or undefined; Yet one can perform the experiment and compute the means that are obtained. What do they look like?
In a different direction: Is there some mathematical quantity that can describe the practical behavior of these experiments, if expectation is not relevant?