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gwern comments on Where Fermi Fails: What is hard to estimate? - Less Wrong
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Counter-challenge: come up with estimating real-world problems, not the mathematical examples.
The rate of change in atmospheric pressure in Moscow exactly 1 year from now in mbar/hour.
Presumably deterministic chaos-based problems, like the famous 3-body problem would qualify, or anything relying on the butterfly effect.
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?