Edit: this was unintentionally redacted!! Is there a undo-the-redaction option?
Sally and John both play a videogame where their characters live on a circle. Starting next to Sally, John walks around the circle 10 to the 100th times (he is a known hacker). Sally hasn't moved and measures the position of John. Assuming that Sally is at the point (1,0) and John moves counter-clockwise, what is the slope of the trajectory of the projectile that John's character fires towards the center of the circle?
For a real one: what is the length of the coast of Britain? Perhaps this is ill-defined without giving a length-scale, but even with a scale (say, a meter) I'm not sure it's doable. Or similarly, what's the surface area of a convenient piece of broccoli?
I have a whimsical challenge for you: come up with problems with numerical solutions that are hard to estimate.
This, like surprisingly many things, originates from a Richard Feynman story:
So what would you ask Richard Feynman to solve? Think of this as the reverse of Fermi Problems.
Number theory may be a rich source of possibilities here; many functions there are wildly fluctuating, require prime factorization and depend upon the exact value of the number rather than it's order of magnitude. For example, I challenge you to compute the largest prime factor of 650238.
(My original example was: "For example, I challenge you to compute the greatest common denominator of 10643 and 15047 without a computer. This problem has the nice advantage of being trivial to make harder to compute - just throw in some extra primes." It has been pointed out that I forgot Euclid's algorithm and have managed to choose about the only number theoretic question that does have an efficient solution.)