You may be looking for the opposite of what you stated; Specifically, there is a game with infinite expected value, but it is not one that you are likely to play: See the St. Petersburg Paradox for an example. Alternatively, in the case of rare events such as winning the lottery, it may make sense for one to play because the positive utility that he/she receives from the excitement may outweigh the negative utility incurred from the financial cost.

My response before was the first idea that came to my mind, but I now want to elaborate with a more thoughtful answer. My general answer is that I don't believe that expected value for rare events can be applied uniformly to individuals. In fact, my point of view extends to certain instances with reasonable chances.

For example, consider a game that is only played *once* which involves a coin flip of a fair coin. If it lands on heads, you win $\$1002$, but if it lands on tails, you lose $\$1000$. The expected value here is of course $+\$1$ so it suggests that you should play. But of course, it is a meaningless value because you either win $\$1002$ or you lose $\$1000$, and that's the end of it. In short, you don't ask what the limit of a convergent sequence $\langle a_i| i \in \mathbb{N}\rangle$ is if you want to compute $a_0$.

Now you can argue against this point by saying that over the course of a lifetime, you will get many opportunities so you should take every favorable one that you can. However, the value of the loss or gain and the number of opportunities that one gets to make bets with such clear-cut odds depends on the individual's situation and preferences. In particular, there's an opportunity cost associated with the bet due to a limit on a person's financial resources.

Similarly if a person is offered two one-time games each yielding the same expected value but with one having greater variance, the choice the individual should make is dependent on the same considerations including the individual's appetite for risk. The expected value from a purely monetary point of view will not differentiate the games.

This captures the spirit of the ideas mentioned in other posts regarding utility and one-time games.

I think expected value is a much more important consideration for large insurance companies because they cover many people, they have a large financial resource pool, and they have a collective utility curve per dollar earned that is much more linear than that of an individual.

As far as cutoffs are concerned, I once proposed that cutoffs of expected value should be in accordance with the number of trials meaning that if the expected number of times an outcome would occur were less than $1/2$ given the number of times that the experiment were to be performed, then it should not be included in the expected value. The problem that was pointed out to me is that if we had a game with a huge number of distinct positive values each with very low probability but having a collective high probability, then this adjusted expected value would be biased in a very misleading way.