Here's another fun example of patenting something impossible: compressing random data. [1] Amusingly, Jean-loup Gailly's [2] post about it begins with saying that: "The US patent office no longer grants patents on perpetual motion machines..."
Every economic proposal that promises a free lunch has a hidden cost, whether the proposer knows it's there or not.
I remember once idly watching one of those "Get Rich Quick Through Real Estate Swindling" seminars on late night TV. Just for fun, I decided to work through the blizzard of transactions the host proposed to see where the extra magic $5,000 it promised to drop in your pocket came from.
Turns out it relied on selling a sucker a bond for $X+5000 that you'd bought for $X, and convincing the sucker that $X+5000 in 10 years was worth $X+5000 today.
There are several classes of problem that really seem to bring out the patent crackpots, of which perpetual motion is the most classic. Apparently data compression is another.
The one I've personally run into is "secure" or "trusted" embedded systems. I've been contacted by a lot of people who believe that they can make a perfectly secure and trustworthy secret storage system just by building it out of an embedded Linux system. When I try to explain how a secure element is basically exactly the opposite of a Linux box, the excuses and bluster prevent me from actually making any progress in explaining.
There's a term for this: "Not even wrong." That is, so off base that "wrong" is generous!
You wouldn't believe how many people on reddit have proved the Collatz conjecture or found a polytime algorithm for 3-SAT...
If a problem is hard enough and stated easily enough, it is crackpot-central.