Here's a simple bit of statistics for a Friday lunchtime. You count the number of galaxies in a certain area on the sky (with the galaxies satisfying some specific properties, if you like). What is the true number density? Let the expected number be $\lambda$ (the true number density multiplied by the area on the sky) and the measured number be $k$. Then, in true Bayesian fashion, what we want is

Now, for the prior, $P(\lambda)$, we assume a prior which is flat on a logarithmic scale. That is, we guess (before making the observation) that the expected number is as likely to lie between 1 and 10 as it is to lie between 1000 and 10,000. (The alternative, a flat prior on a linear scale, would mean that we guess the true density is just as likely to lie between 10,001 and 10,010 as it is to lie between 1 and 10, which is ridiculous.) So $P(\lambda) \propto 1/\lambda$. The likelihood, $P(k|\lambda)$ is given by the Poisson distribution. So, ignoring the normalizing factor of $P(k)$,

And this is the Gamma distribution. Easy peasy.