My understanding of the differences between MH and Gibbs Samplers is that a Gibbs Sampler is usually used when the full conditionals are present to us. In other words, it is a known distribution, so that sampling from it is just as easy as calling a function in R.

I would usually use MH if the conditionals are not well known. However, I am failing to think of an example which cannot be done by Gibbs and must be handled by MH. Is there a prototypical hierarchical model? Thanks.