Yet another challenge for you: how many times has the water you drink been pissed out of a vertebrate (something with a spine) in the past? If the number is very small, then those who baulk at drinking recycled water have more cause to complain than if the number is very high.
The question is quite relevant to Australia: within Australia there is a large public antipathy towards recycling waste water because of the ‘yuk’ factor whereby people don’t like the idea they are drinking reconstituted piss. This reluctance has real consequences for urban water shortages. In other countries they are not so fussy, so it matters whether or not water has been piss many times already. If not, the ‘yuk’ factor is more understandable. If, on the other hand, water has been piss many times in the past already, the ‘yuk’ factor is a matter of ignorance in that one cannot avoid drinking reconstituted piss.
I can already say that there is no official answer to this one, at least not that I have found, so you will be breaking new scientific ground with a good estimate. I thus invite you in the comment thread to give a decently founded guess.
So, what’s the question exactly? The question is how often the atoms in the average water molecule will have been part of the piss coming out of a vertebrate. This includes all the mammals and fish and dinosaurs and what-have-you that ever inhabited the earth, but it excludes things like bacteria and krill. And we are talking about piss here, so sweat is not included nor water sifting through the gills of fish or passing through the mouths of whales. If we would include those latter forms of water that have also passed through bodies of vertebrates, the answer is bound to be hundreds of times as high. We are also talking about ‘human-like’ piss, meaning that if it is diluted or concentrated relative to us, we are looking for human-equivalence terms.
My best guess next Monday.
 Why the use of the word ‘atom’ here? Well, because it turns out that water molecules continuously break up and mingle their atoms with other molecules. Hence your average water molecule isn’t stable or all that old. But the atoms making up a water molecule will not be new and not be subject to continuous break-down and rejuvenation, at least if we stick to the usual convention of not counting the sharing and pass-through of electrons as somehow creating new atoms.