Considering that bitcoin addresses get wasted and discarded like plastic bottles of water, is there ever a risk that we'd use up all the bitcoin addresses?
As others pointed out: not really.
However, the possible number of addresses is not really 2^160, but rather 2^80, and in reality much smaller. The reason for that is what is called the "birthday paradox". If you chose 2^80 different addresses at random, out of 2^160, then you start to have a significant probability (several 10%) that two of them are equal. Which would mean that the private key of the one, also works on the other one. To make this probability small, one should be a huge factor below 2^80. Say, 2^50 or so. That's still of the order of one million billion addresses.
So it is reasonable to limit each human to a few tens of thousands of bitcoin addresses if we want bitcoin to last for a few generations.This is in fact more limiting than one might think, if one also wants to adhere to "no address re-usage". It would mean that each of your transactions, in your whole life, would need a new address. Now that starts to be problematic, because if you do 10 transactions a day, you do 3000 a year, and in 3 years time, you've used up all of your address quota.
But then, we could allow for 2^60 addresses. That would increase slightly the "collision probability", but it would still be very small (of the order of 1/2^40). At that point, each human would be entitled to 1000 times more addresses, which would solve the issue.
