Why would we ever need 10k new addresses in average per day? Generally speaking, the address space of Bitcoin seems to be sufficiently large to make it unlikely that a hash collision will ever occur (see below).
Pacerier stated in a comment on Is it possible to brute force bitcoin address creation in order to steal money? that we'd "want Bitcoin to scale to [...] 10b people, each generating 10k different addresses a day, that's 100 trillion addresses created daily." He followed up with "10k may be a severe underestimation".

100 trillion is 10^14, so if we kept up that pace, we'd have used one address per satoshi after 21 days. I am confused as to what might be the reasons for users to need 10k addresses per day.

Please explain: Why does it seem likely that we will need such a large number of new addresses each day? In what scenario would each user in average need 10k new addresses per day?



Total number of satoshis:


At one address for each satoshi that could be ever in existence (2.1*10^15), with the giant address space of 2^160 possible addresses, the simple approximation of the Birthday paradoxon, which works well for probabilities less than or equal to 0.5...



... tells us that the chance of single hash collision occurring is still pretty small: 1.508*10^(-18).
Bitcoin transactions usually carry change, and to increase privacy and security, it is certainly no worse to generate a new address every time for receiving change than to reuse old addresses. Details maybe belong into their own question; however, it is well within the realm of reason to expect a future where most Bitcoin transactions include two new addresses (one for the person paid and one for the change back to the payer).

If we dream up a future where arbitrarily small micropayments can be handled by Bitcoin, then I would actually call the 5k transactions per user that could hence generate 10k new Bitcoin addresses a vast understatement of the potential. I'm thinking that rather than paying my mobile carrier for data roaming, I'd want to use micropayments to pay for routing of IP packets in adhoc wifi networks (ok, maybe not for each individual packet given the overhead---but I could, from a user's perspective, want that to happen for every 10 kb chunk of data or so). And I can envision much the same thing with my regular ISP, or even with most websites I frequent (and, come to think about it, maybe some of them should pay me).

The problem with this is that all these transactions end up in the blockchain, occupying a huge amount of disk space. That is why many suggest that micropayments ultimately belong off-blockchain, meaning some other solution should be found for them that does not place the burden on Bitcoin. In my example, it is easy to see why this is necessary: Even if only 100 nodes kept the entire blockchain, then my 10 kb chunk of transmission data would occupy much more than that in distributed blockchain storage just to finalize my payment for it. Obviously not just inefficient but truly unsustainable at scale.

Hence I would say that as far as Bitcoin's blockchain is concerned, we need not worry about seeing quite that many addresses added to it. But if whatever off-blockchain solutions we will adopt for micropayments in the future uses Bitcoins' address scheme as well, then we may well, outside the blockchain, envision such a massive use of Bitcoin addresses.