I doubt if much compression is possible of much of the chain. Compression is possible where data exists in non-random patterns such as words from a human language. If you created a paragraph of truly random characters no compression would be possible without loss of data.

In the case of a blockchain a lot of it is in the form of cryptographic hashes that are pseudorandom in nature. In principle I think they would be incompressible. Other portions could probably be compressed a bit, but I'm not sure it would be worth it. And if it is worth it, I bet it has already been done.

The developers have done a _lot_ of work optimizing Bitcoin code (far more than is generally appreciated, due to the endless lies and FUD hurled against them). So if there was any obvious compression work to be done I'm sure it's already been implemented. If you look at techniques like the recently announced reduction in size of CT (confidential transactions) you'll see this is high on their list of priorities.