Re: There are 2^256 private keys out there: how big is that number?
that is technically incorrect.
they are not exactly finding private keys with balance, they are solving a puzzle. a long time ago (2015) in order to show the hugeness of the private key space (or maybe just for fun) someone created a "puzzle" where he chose keys in a certain smaller space and sent increasing amounts to each of those keys like this:
201 send 0.001BTC=$0.2 at the time
212 send 0.002BTC=$0.4 at the time
223 send 0.003BTC=$0.6 at the time
and so on.
now, people to this day are still trying to solve that puzzle. so technically if you have a private key (which is impossible by the way) that is in one of those ranges they won't find that because they are only looking to solve that puzzle.
In the other hand they somewhat evolved, as they claim they are looking for the whole 2^160 addresses.
p.s. a bit nitpick-y but there are 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEBAAEDCE6AF48A03BBFD25E8CD0364140 private keys which is a little smaller than 2256 
I know, actually the number reported in the post is not 2^256, but the decimal equivalent of 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEBAAEDCE6AF48A03BBFD25E8CD0364140
I doublechecked with this message.
Anyway I edited the OP for clarity. Thanks