What were some back-of-the-envelope estimates that scientists made where the result of the estimate was important to either the intellectual development of the field or had significant social consequences?
For example, Helmholtz made some simple estimates to show that chemical energy could not possibly fuel the sun for its known lifespan, then argued that the only remaining energy source was gravitational, and used this to estimate an (incorrect) upper bound for the age of the sun. A detailed model or extensive calculation wasn't necessary; his points could be made just as well with order of magnitude numbers.
This is different from, for example, Fermi's famous estimation of the yield of the first atomic weapon test by dropping slips of paper. While his estimation was impressive, there was (I presume) a team of scientists collecting far more data and crunching through more detailed models to calculate the yield to much greater accuracy than Fermi a short time later, so while his estimation was interesting and amusing it had little real consequence. On the other hand, Taylor's estimate of the Trinity test yield based on photos from a magazine might count because he was able to reveal important classified information with this estimate, and he came to with a very simple method, not needing the more detailed approaches he had been considering.