I think we agree then:

- for a fixed sample size we can find a time large enough that the probability of the sample mean being above $1 is as low as we want

- for a fixed time we can find a sample size large enough that the probability of the sample mean being below $1 is as low as we want

- when both the sample size and the horizon grow without limit which effect dominates will depend on how we make it happen

Adding "almost surely" to "everyone goes bankrupt and will never recover" or "there is a finite time after which no-one ever passes above $0.0000000000000001" is a subtle change but it's enough to allow for someone to go to infinity with infinitesimal probability.

This is why the distribution mean can grow exponentially, it wouldn't be possible if the everyone and no-one in those quotes were strictly true.