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I think more about stamps where you can see the damage directly in the photo without reading the description. If you know plenty of them, I will be glad to see them here in this topic, of course.
Take a look at the Siegel census page of rare US stamps. Some examples of the rarities have faults, and most of them are illustrated on the census page for their issue:
https://siegelauctions.com/census.phpThis includes past auction realizations, so you can also compare how much the faulty rarities realized in sale.
The Large Persian Rug (#R133) is a famous US rarity that is almost always faulty, although the most gruesome-looking copies generally were repaired decades ago.
They aren't really
famous rarities, but if you head into the deep back of the book you'll find a few more. There are a number of 19th century carrier stamps, postmaster's official seals, postmaster provisionals, and revenues that are immensely rare (say, ten or fewer copies known) and always or almost always faulty. They aren't too frequently seen though.