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 be able to rule out 1 of 2 hypotheses: 100% heads and 100% tails.

But this isn't just true of the first observation. Suppose I flip twice, and get heads then tails; so the single most likely frequency is 1/2 since that's what I have to date. But now we're back to the same situation as in the beginning: we've managed to accumulative evidence against the most extreme biases like 99% heads, so we have learned something from the 2 flips, but we're back in the same situation where we expect the posterior to differ from the prior in 2 specific directions but cannot update the prior: the next flip I will either get 2/3 or 1/3 heads. Hence, I can tell you - even before flipping - that 1/2 must be dethroned in favor of 1/3 or 2/3!