I have been building a few of my own MCMC algorithms for hierarchical Bayesian models. If the posterior distribution of say $\alpha$ is analytically tractable, I sample $\alpha$ using an R function such as rgamma with the correct parameters. If the posterior for some parameter, say $\beta$, is analytically intractable, I use a Metropolis-Hastings ratio. Throughout the first seven algorithms that I've built, I've noticed that every time a parameter has an analytically tractable posterior distribution, it's in a conjugate relationship; every time I need to use an M-H ratio, it is not a conjugate.
Now, I know that conjugacy makes determining the posterior much easier, but are there ever times when one cannot analytically derive the posterior of a conjugate prior in some hierarchical model (i.e., when using latent indicator variables)? Additionally, are there occurrences where I can analytically derive a posterior distribution that is not in a conjugate relationship?
One further question, are there other relationships which always result in a known posterior distribution that is not conjugate? (Fake Example: We use a Binomial prior and the data follows a Poisson distribution. Then the distribution will always be a $\chi^2$ distribution.)