As detailed here and elsewhere, Feynman and others at Los Alamos could calculate many problems to 10% accuracy in minutes:
When I was at Los Alamos I found out that Hans Bethe was absolutely topnotch at calculating... A few minutes later we need to take the cube root of $2 \frac 1 2$... and he says, "It's about $1.35$."... I had a lot of fun trying to do arithmetic fast, by tricks, with Hans... I announced, "I can work out in sixty seconds the answer to any problem that anybody can state in ten seconds, to 10 percent!"... People started giving me problems they thought were difficult, such as integrating a function like $\frac 1 {1 + x^4}$, which hardly changed over the range they gave me. The hardest one somebody gave me was the binomial coefficient of $x^{10}$ in $(1 + x)^{20}$; I got that just in time.... [Paul Olum] says, $\tan 10^{100}$. I was sunk: you have to divide by $\pi$ to $100$ decimal places!
What methods do Feynman, Bethe, Olum, use to do these? Or, since we can't really know the answer to that: What methods can we use to easily approximate calculations within 10% error?
Now, one might simply respond: Wolfram Alpha. But we do this not for lack of a calculator! For example, Sanjay Mahajan requires his students to show "number sense" by
Without a calculator, estimate $\sqrt{1.3}, \sqrt[3]{1.6}, \sin 7,$ and $1.01^{100}$.
What methods can we use to do this? I'll post my collection as an answer below. Surprisingly, while I've found good methods for estimating logs, exponents, and trig, approximate long division (in less steps than the real thing) proves to be the hardest!