The halting problem only applies to full Turing machines, and their equivalents. So, a compression algorithm based on x86 assembly is uncheckable, but one based on a restricted format like .zip would be fine. Or, you could just give it a time limit.

I made the file DanBC referred to above, and wondered about how much computation you could do with iterated decompression (specifically, deflate-type decompression). Could you make a zip file representing a Turing machine, so that after unzipping it it represented the machine after one step of processing, and after unzipping the result, represented the machine after a second step, etc.

Or if not a direct translation of a Turing machine, then some other Turing complete system.

As far as I know, no one has done this, but if it is possible, one implication would be that you couldn't prove whether the process of recursively unzipping a file could continue forever, or would halt.