There are tools like EuroBillTracker (https://en.eurobilltracker.com/) where you can enter the serial number of bills that you received and can watch them travel around the world. If someone else tracks them as well, that is.
I entered 31 serial numbers over the last 15 years. Those haven't been seen again so far. I guess it's not the right kind of game for me... :)
If there were money in it, someone would throw OCR at the problem. Say, attach prizes to certain bills, or finding certain patterns of bills (say, two bills whose serial numbers are mathematically related a certain way).
You can still play that game with new coins, but it’s less f visible now, as most coins are old and those already are well dispersed throughout the euro zone (and, of course, more and more people pay with a card)