"A blind use of tail-area probabilities allows the statistician  to cheat, by claiming at a suitable point in a sequential  experiment that he has a train to catch. This must have been known to Khintchine when he proved in 1924 that, in sequential binomial sampling, a "sigmage" of nearly sqrt(2 log(log n)) is reached infinitely often, with probability 1. (Weaker results had been proved earlier by other mathematicians.) But note that the iterated logarithm increases with fabulous slowness, so that this particular objection to the use of tail-area probabilities is theoretical rather than practical. To be reasonably sure of getting 3 sigma one would need to go sampling for billions of years, by which time there might not be any trains to catch."

I. J. Good, in the discussion of J. W. Pratt, "Bayesian interpretation of standard inference statements,"  J. R. Statist. Soc. B 27, 169 (1965), page 196.

Many thanks to Luc Demortier for digging this out and sharing it!