Multiplication of normal distributions
The distribution of the multiplication of independent Gaussians is not close to a Gaussian.

There are various ways to convince yourself in this. E.g., compute the expectation and the variance of the product from the expectation and the variance of the individual variables.
This gives you a candidate Gaussian if there were a Gaussian approximation.
Now, think of some simple properties of the product (e.g., the probability it is 0) and see if your candidate is good enough as an approximation.