My favourite was an old p2p network - I think it was called OFF? It never caught on, but it had a very interesting concept.
Let's say you have a copyright-infringing file - call it Bieber-generic-love-song.mp3. You don't share it directly. When you put it into OFF, the client will see if it has a block of data of matching size or a little larger. If not, it'll create one - full of completely random bits:
Now, it takes your Bieber-generic-love-song.mp3 and XORs that with Garbage1.bin. That gives you Garbage2.bin... which is also purely random, because it's a result of an XOR with uncorrelated random bits.
Now you have two chunks of data, Garbage1.bin and Garbage2.bin, both of which are utterly random - they can't possibly be infringing upon copyright in any way, because they contain no meaningful information. But the network also has a search function - and if someone were to search for Bieber, your client would answer: "I know of Bieber-generic-love-long.mp3. To get it, use Garbage1.bin and Garbage2.bin, truncate to X bytes."
The searcher than goes and downloads Garbage1.bin and Garbage2.bin - both of which are, on their own, nothing but random bits. And from those, through the magic of mathematics, out pops the latest vapid ode to an unnamed girl from a manufactured pop star.
The overhead is bad - up to 100%.
It never really caught on because of the overhead and because better, though more legally-dangerous, networks also existed. But it shows an interesting approach to using mathematical trickery to subvert the law. Somehow I doubt it would stand up in court - judges tend to frown upon people who find creative ways to avoid infringing the letter of the law while making an obvious mockery of the intent.
The wikipedia page still exists, but the website of the software doesn't.
2006 was a time of great optimism for the pirate community - I was in university at the time. Napster had been shut down, but countless successors were blooming and it really felt like we would bring down 'The Man' and usher in a new age of free access to knowledge and unconstrained international flow of communication. The future felt inevitable. Turns out we were wrong. I wonder if this is what the hippies felt like as they grew older, realised the flaws in their youthful vision and watched their movement fade.