Other than Normal prior Let $X\sim\mathcal{N}(\mu, \sigma)$, where $\sigma$ is known, If the prior for $\mu$ is normal then the posterior for $\mu$ is also normal. My question: Is there any other prior for $\mu$ that makes the posterior of $\mu$ normal?
 A: I'll assume you only want proper priors.
Given that $L(\mu)$ (the likelihood) is proportional to a Gaussian, you're asking for a prior density $f(\mu)$ such that the unnormalized posterior $p(\mu)=f(\mu)L(\mu)$ is also proportional to a Gaussian.
That's the same as saying you want a $f$ such that $\log(p)=\log(f)+\log(L)$ is quadratic, where $\log(L(\mu))$ is itself quadratic.
Clearly, then, as the difference of two quadratic functions, $\log(p)-\log(L)$ (i.e. $\log(f)$) will be quadratic, linear or constant. 
If you constrain $\mu$ to lie in some bounded or semi-bounded range (so that constant and linear polynomials could also work as log-densities), you end up with a posterior which is normal but truncated to lie within those bounds. [I assume you don't consider that actually normal.]
The only remaining informative possibility is that $\log(f)$ has quadratic leading term - and since we're on the real line, only with negative coefficient (since it has to have a finite integral over the line so we can scale to a density). After scaling $f$ to integrate to 1, this is the normal family.
A: Yes, the improper prior for $\mu$, i.e. $p(\mu)\propto 1$, results in a posterior for $\mu$ that is normal.
