Are there analytically derivable posteriors that save from doing MCMC other than conjugate priors? Posteriors for conjugate priors can be analytically derived and save us from doing MCMC.
Conjugate priors simply have a posterior in the same family as the prior distribution. Are there other analytically derivable posteriors that save us from doing MCMC?
 A: Anytime the posterior distribution is a known, well studied distribution, chances are we can sample from it. Some prior and likelihood combinations lead to a posterior that is in the same class of distributions as the prior. When this happens, the prior is said to be conjugate for the likelihood.
There are a wide variety of such prior-likelihood combinations like the Normal-Normal, Beta-Binomial etc. A complete list can be found here.
There can be other examples of when the posterior distribution is known, but not necessarily in the same family as the prior. 
$$ X|\sigma^2 \sim N(0, \sigma^2) \quad \text{and} \quad \pi(\sigma^2) \propto \dfrac{1}{\sigma^2}.$$
The prior on $\sigma^2$ is improper (and if the parameters in the Inverse-Gamma were allowed to be 0, this would be an Inverse-Gamma distribution, but it is not).
The posterior distribution is
$$\sigma^2|X \sim \text{Inverse Gamma}\left(\dfrac{n}{2}, \dfrac{x^2}{2} \right).$$
So, yes there can be other examples of the posterior distribution taking a known form, such that MCMC methods are not needed. In addition, even if the posterior distribution is unknown, it is not necessary that an MCMC procedure is required to sample from the posterior. Rejection sampling may be implemented. For example,
$$X|\theta \sim N(\theta, 1) \quad \text{and} \quad \theta \sim \text{Cauchy}(\mu, \sigma).$$
The resulting posterior is intractable, but a rejection sampler can be implemented to sample from the posterior (I can elaborate on how this is done, but that would be digressing).
