That seems like another way of wording what I said, which is that there is no way to predict the output other than by running the algorithm that generates it. I

That seems like another way of wording what I said, which is that there is no way to predict the output other than by running the algorithm that generates it. If you are presented only with the string representing the unbounded output, is there any way to reconstruct the algorithm from it? If not, then I would say that the string itself is incompressible; in that case only the string plus the algorithm is compressible in the way you describe.