Is it illegal to charge insane prices for things?

Nope. That's why the rich use art as a way to launder money. Set the price you want to launder and no one bats an eye.
- Like NFTs!

If it can be proven to be "self dealing" aka your NGO, charity, or non profit purchases or accepts a tax deductible donation of say, a portrait of you, from you or one of your businesses for hypothetically, 100 million dollars, then you may, if you did something particularly odious such as say, run for president. Then you might lose the legal ability to run that sort of organization.
- This is the correct answer. Overcharging (and self-dealing) is typically used in tax fraud and money laundering.
  Of course, no one would actually do that, especially if they were to later do something as public as running for president. The entire justice system, due to its fair and impartial nature, would come crashing down on their head with every resource at its disposal, and the people would riot in the streets if it treated such egregious crimes as less important than passing a bad $20 bill.

I don’t think there’s any Federal laws against it, but the closest I can think of is that in many states there’s laws against price gouging in some situations (after a state of emergency has been declared, may only apply to certain essential goods/services), here’s a FAQ on California’s anti-price gouging laws:
https://oag.ca.gov/consumers/pricegougingduringdisasters

In certain circumstances you're not even allowed to sell things for more if someone OFFERS it! For example, if you have pallets and pallets of bottled water, and a hurricane just tore through, all of a sudden your water is much more valuable. You could conceivably sell it all for 10 or 20 times the price.
But that's called price gouging, it's illegal, and it's something that only total potatoes do. It's also something that might get you hurt later, once things settle down and people remember that you were trying to charge them $20 for a bottle of water when there was a boil water order on.

No, but if the sale goes through and you don't declare it to the tax man then that's probably illegal.

There are laws preventing someone from jacking up the prices of necessary items during emergencies. But I think the law of supply and demand is the only thing stopping you from selling a baseball hat for $20,000.

No, not really. If there was some other goal besides the honest exchange of the hat for the money, then maybe. Fraud ("the hats made of gold and is therefore a great value!") and price gouging ("it's the only hat for sale on a sunny day so I'm selling it at a 1000x markup!") might be crimes in some contexts. As mentioned, money laundering (avoiding tax or other legal requirements to move money) is also a crime.

No.
Even Martin Skreli, the dipshit who jacked up insulin prices by 300% didn't get in trouble for that; he went to prison for security fraud or someshit.
However, if there was say a crisis going on and you raised the price of water you were selling to capitalize on others' misfortune that tends to be illegal. Like when stores were price gouging on hand sanitizer during the height of the covid lockdowns.
- Skreli was absolutely 100% capitalizing on others' misfortune and the fact that he didn't get in trouble for that is incredibly bad.

In addition to what others have posted, as long as you're honest about it just being a hat and not making any special claims about it (like it was owned or worn by a famous person), I think you're good.

That depends entirely on the country you're in, and the thing you're trying to sell.