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Examples where Kolmogorov's zero-one law gives probability 0 or 1 but hard to determine which?

Inspired by this question, I was curious about a comment in this article:

In many situations, it can be easy to apply Kolmogorov's zero-one law to show that some event has probability 0 or 1, but surprisingly hard to determine which of these two extreme values is the correct one.

Could someone provide an example?

Jason Dyer
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