How many Bitcoins have been provably lost? At least ...

[DannyHamilton]

Does a miner have choice for his reward ?

Yes.

The protocol allows the miner to claim any reward equal to or less than the sum of the block subsidy and all the transaction fees of all the transactions included in the block.

[zebedee]

Another source of lost coins is due to miners taking less than the maximum block reward which in effect "de-mines" an amount of coins equal to the difference between the allowed reward and the taken reward.

I'm aware of two coinbases that have identical hash to two others.  This represents a permanent destruction of 100 BTC.

https://blockchain.info/tx/e3bf3d07d4b0375638d5f1db5255fe07ba2c4cb067cd81b84ee974b6585fb468
https://blockchain.info/tx/d5d27987d2a3dfc724e359870c6644b40e497bdc0589a033220fe15429d88599

This is why version 2 blocks embedding block height were created.

What I'm curious about, and it's hard to check without software, is if either of the earlier in time of each of the two was spent before the later one was created.  In which case the coins wouldn't be lost.  I suspect they were unspent and lost, but it'd be cool if someone confirmed.

[teukon]

As of block 305,303 the coin supply is limited to 12,882,575 BTC.

I think this should be 12,882,550 BTC (or 12,882,600 BTC if including the genesis block reward).

[teukon]

Back on topic.  D&T are you asking us to supply other known amounts?  If so you should add the 1.8252962 at https://blockchain.info/address/1BitcoinEaterAddressDontSendf59kuE

To prove that these bitcoins are invalid one would have to prove that there is no corresponding private key and I suspect that this is a hard problem.