They don't even have to be fat-tailed; in very simple examples you can know that on the next observation, your posterior will either be greater or lesser but not the same.

Here's an example: flipping a biased coin in a beta distribution with a uniform prior, and trying to infer the bias/frequency. Obviously, when I flip the coin, I will either get a heads or a tails, so I know after my first flip, my posterior will either favor heads or tails, but not remain unchanged! There is no landing-on-its-edge intermediate 0.5 coin. Indeed, I know in advance I will b... (read more)