This question is related to Laplace's demon and the solar system as a chaotic system, as well as the paradox with the hare and the turtle (I think).
Imagine you want to calculate the future of the solar system. Imagine also that the distance between two planets is $\sqrt{2} + \sqrt{3}$. Since the solar system is a chaotic system, you'd need infinite precision. Even if you're an all-powerful demon, would it be possible to do the calculation?
This boils down to: what do you mean by "add"? Given two finite expressions representing real numbers, such as $\sqrt{2}$ and $\sqrt{3}$, you can add them in finite time by simply putting a $+$ between them: $\sqrt{2} + \sqrt{3}$.
If you mean you want a numerical representation with infinitely many decimal places, of course you can't do that in finite time unless you can write down an infinite output in finite time. I'm not familiar enough with "all-powerful demons" to know whether they can do that.