I keep seeing references to destroying bitcoins by sending them to addresses that nobody can own (e.g., addresses derived from impossible public keys). What would be the reason for doing this? It's discussed in such a cavalier manner in various posts that I'm assuming it's not an attempt to undermine the currency.
Certain escrow techniques rely on one party being able to "destroy" these bitcoins, without being able to steal them. For example I could put 10 bitcoins in escrow, giving a second person the ability to destroy them. Now if that second party lends me, say, 5 bitcoins, I can't simply run away from my debts without incurring a loss larger then the debt itself, once the other party destroys the bitcoins. There are variations of this scheme for a number of other purposes.
You don't want to actually destroy bitcoins in such a scheme, but you want the threat of destroying them.
But "destroy" in this context isn't necessarily an actual destruction, as long as "destroyed" bitcoins are a loss for both parties. Donations to trusted organizations are an alternative.
A non-malicious reason to deliberately destroy notable amounts of Bitcoins would be for a hypothetical alternative blockchain that recognizes Bitcoins sent to a special address as newly generated coins. One could for example try to establish a new "Altcoin" which is incompatible to Bitcoin, but every private key that signs a Bitcoin transaction to a special impossible Bitcoin address is eligible to also spend an according amount of Altcoins. This would enable a secure one-way conversion to a separate blockchain without the need for exchanges.
I don't think anyone is intentionally destroying large amounts of Bitcoin. Some relatively large amounts have been destroyed accidentally, due to hardware and software failures. This thread is an attempt to count all the lost Bitcoins.
Don't forget that each of the 21 million total Bitcoins is divisible into 100 million parts. So it's considered that there's plenty to go around, and the more that is destroyed, the more the remaining ones will be worth.
Also, don't confuse the term "bitcoin days destroyed" which is sometimes used with destroying the coins themselves. This "days destroyed" measure is sometimes used as a measure of velocity in the bitcoin economy, and doesn't really measure destruction of anything. If you have 10 bitcoins for 7 days in the same place, we can say they have an age of "70 bitcoin days". If you spend them, the new coins will have an age which starts again from zero.
There have been at least one message services / blockchain tattooing services that will make an unredeemable(?) TX to a vanity address.
If one were to have enough hashing power and time, theoretically you could recover keys for lost or destroyed coins. A lot of time. ;)