My Favorite Strange Number: Ω

The bit in parentheses is why your argument doesn't show that omega isn't incompressible: it very emphatically isn't computable.

you're right that the isolated assertion is not at all sensible, but the wider context is that you are considering all possible algorithms and the sequences that they generate, and looking for some way of compressing as many of those as possible. you should read one of chaitin's books for a better idea (they're generally "accessible", but at some point you just give up trying to make complete sense of it all...)

in other words, you are choosing your compression for one particular algorithm (the one that generates omega - you could go further and say it's 1 for omega and 0 for everything else). this is a common problem in arguments about compressibility (if we're only trying to compress beethoven's ninth then i can do that in one bit too) - you have to be talking in an "average" or statistical sense for things to be useful.

having said that, the exact nature of what they're saying is still not clear to me (ie it's not clear exactly how the details of the argument work out, or how this eventually ties back to http://en.wikipedia.org/wiki/Shannons_source_coding_theorem).

That's not quite fair because you have to include the size of your decompression routine with the description.

Also, there's the whole "not computable" part.

When he says "compress," he means "losslessly." A compression which doesn't actually allow you to compute any of the original data is pretty lossy.