(Clearly not a pardox per-se but I would like to hear what you think)
The basic riddle (not a very interesting one even) goes as follows:
A first client comes into a barber shop, takes a hair cut and the barber tells him "check how much money there's inside the cash box, double it and take $\$20$ change back.
A second client comes, takes a hair cut and the barber tells him "check how much money there's inside the cash box, double it and take $\$20$ change back.
Same story with the third client, who happens to be the last one.
At the end of the day the barber checks the box and it's empty. How much money was in the box at the beginning of the day?
So, it's not too difficult to compute and to get that the answer is $\$17.5$.
Generalizing for $N$ people, and for $K$ amount of change each time, we get that in the beginning of the day there was
$$K \cdot \frac{2^N -1}{2^N}$$ initial amount in the box.
Now, what happens when there are infinitely many clients? The limit of the above sequence is clearly $K$ but on the other hand that would mean that even though each client is doubling the amount in the box and taking half of it, after infinitely many clients the box is empty!
I think that the "paradox" here lies within the question itself. You can't talk about what happens after infinitely many clients.
Am I right?