Based on Andrey's 2^160 addresses, I did a thought experiment.
Let's say Dr. Evil wants to occupy all the bitcoin addresses, because of something Satoshi once said about his ears.
He thinks of using 1,000 evil computers for the task, but he's in a hurry and he's quite wealthy, so he buys 1 million evil computers instead.
He sets each of those computers to generate not 1 address per second, but 10,000 addresses per second.
Dr. Evil sets his evil botnet running and sits back in his chair laughing evilly. But the last laugh is on him.
totalAddresses = 2 ** 160
secondsInOneYear = 60 * 60 * 24 * 365.25
addressesPerSecond = 1000000 * 10000
yearsTaken = totalAddresses / addressesPerSecond / secondsInOneYear
It will take 4.6 x 10^30 years for his bots to occupy all the addresses.
(That is assuming his computers always generate unique addresses.)
If, instead of one Dr. Evil trying to carry out this nefarious plan, one million Dr. Evils coordinate their efforts to reach the same goal, the exponent will fall from 30 to 24.