in a series (one single, continuing series!) of coin tosses, the probability that you get a run of heads at least half as long as the overall length of the series (eg ttththtHHHHHHH) is always >0, but it is not guaranteed to happen, no matter how many chances you give it.

... any event for which you don't change the epsilon such that the sum becomes a convergent series. Or any process with a Markov property. Or any event with a fixed epsilon >0.

That should cover round about any relevant event.

(and also you mis-apply the Law of large Numbers here)

Explain.