Concentration inequalities for product of gaussians Are there any concentration inequalities (i.e. probability bounds on how a random variable deviates from its expectation) for the product of $n$ independent gaussian random variables with zero means and equal variances? What about different means and variances?

This paper computes the desired probability explicitly in Theorem 3 (page 4).
The theorem states that for independent standard normal Gaussians random variables $g_i$ one has 
$$
\mathbb{P}(X \le t) = \mathbb{P}(X \ge -t) = f_{0.5, 1}\bigg(\frac{2^n} t\bigg),
$$
where 
$$
f_{\nu, \xi}(u) = \nu + \frac 1 {2^{\xi}} \cdot \frac 1 {\pi^{n/2}} \sum_{k=0}^\infty u^{-0.5 - k} \sum_{j=0}^{n-1} H_{kj} \cdot [\log u]^j,
$$
for details and definition of $H_{kj}$ see the paper itself.
The expression seems to be complicate enough to argue to the concentration bound immediately from here, but I believe this is a good starting point. 
