Since I was first introduced to it, I've been intrigued by the claim that the universe contains a finite amount of information. (That link is not where I first encountered the concept; it is simply the first example of this claim I could find from a quick Google search.)
Basically, the argument seems to be that if there is a finite amount of matter in the universe, that matter can only store a finite amount of information. On the surface, I have to concede that this makes a lot of sense. After all, if I'm thinking in terms of bits (for example), I might visualize a hypothetical "infinite hard disk drive" that could store unlimited data. This device would presumably have to be infinite in size, since it stores information on a physical platter that obviously occupies some space.
Digging a little deeper, however, I start to doubt this presumption. After all, information can be compressed according to a system of encoding information in a particular set of symbols. Then as long as the system provides a way of decoding that information, you could effectively increase the capacity of any storage mechanism by encoding its contents using said system (analogous to converting every file on a hard disk using some compression algorithm such as LZMA).
But, there's still more to it than that. It goes without saying that any system of compression like what I just described comprises its own information, and therefore needs to be stored somewhere itself.
Since the universe is "all there is" (?), a system of encoding the information contained within it would have to be a part of that very information. This is where I think I hit a mental wall. On the one hand, it seems that you could extract a seemingly unlimited amount of information from finite data—by using a system to encode that data, another system to encode the encoded data, and so on and so forth—whereas on the other, intuition tells me that there must come a point where, if the data as well as the system must share a space, there is no longer any room for either more data or another system of encoding it. The available space becomes too "crowded," so to speak.
Is there a mathematical principle or theorem that answers this question? Is the problem I'm describing (determining a limit on the capacity of material data to store information) defined, analyzed, and/or illuminated by any particular concept(s) in mathematics?