https://bitcoin.stackexchange.com/a/24070

If you prefer diagrams: https://i.sstatic.net/VWwJq.png (there are more addresses in the address space than there are zeptometres, 1/1 000 000 000 000 000 000 of a metre, in the universe's width).

If you prefer maths: http://download.wpsoftware.net/bitcoin-birthday.pdf (by Andrew Poelstra) says (slightly edited):

Using [birthday attack maths], we calculated [above] that for a 0.1% probability of collision, we would need 5.4 × 10^22 addresses in existence. For a 99.9999% chance, we would need 6.35 × 10^24 addresses.

So, even if there were 10^22 bitcoin addresses generated, a collision simply will not happen. But if there were 10^25 addresses generated, a collision absolutely would happen.

Should we worry about this? No, for these independent reasons:

The chance of getting a specific collision, say, a collision with one of your addresses, is still 1 in 2^160 or 1 in 10^48 . So even if you've got a million million million addresses, nobody has a chance of colliding with you.

At the time of this writing, there are less than 10^7 addresses in use in the network. So anyone with 10^25 addresses would only be colliding their own addresses.

Each address takes around 100 bytes to store. (Actually about half that, but we only care about orders of magnitude.) So for the network to support 10^25 addresses, it would take 10 million million terabytes of storage just to record them. (And this is not even touching the problem of searching such a huge data store.

According to sipa, if the current mining network (which is at 25 THash, and the most powerful computing network in the history of the world) were switched over to address generation, the network could generate 2.5 × 10^12 addresses per second (one address generation corresponding to roughly 10 hashes). At that rate, it would take 127,000 years to get so many addresses. It is debatable whether homo sapiens has walked the earth for that long.

With 21 million bitcoins ever existing, and 8 decimal places of divisibility, at most 2.1 × 10^14 can possibly have money on them at once. But in a space of 10^24 addresses, this means that only one in 10^12 addresses could possibly have money on them. So an attacker, after doing the physically impossible 3 trillion times over, has only a one in a trillion chance of getting even one satoshi out of it.