Is it possible to apply the law of the iterated logarithm (e.g. http://en.m.wikipedia.org/wiki/Law_of_the_iterated_logarithm) to derive non-trivial (i.e. bounded) 100% confidence intervals for population averages?
An abstract (http://www.tandfonline.com/doi/pdf/10.1080/10485250410001713963) gives at least such hint. However, its first reference by Robbins (http://projecteuclid.org/euclid.aoms/1177696786) does not seem to cover such result, as pointed out in @whuber's comment.
Edit: After justified comments by @whuber, I reformulated the question.