I think by the time we reach some future extreme of data density, it will be in a method of storage beyond our current understanding. It will be measured in coordinates or atoms or fractions of a dimension that we nullify.

This is a really good question!

I believe the general answer is, until the compressed file is indistinguishable from randomness. At that point there is no more redundant information left to compress. Like you said, the 'information content' of a message can be measured.

(Note that there are ways to get a file to look like randomness that don't compress it)