What is the equivalent smoothing function to running the same Gaussian 8 times?
From Wikipedia:


  Applying multiple, successive gaussian blurs to an image has the same effect as applying a single, larger gaussian blur, whose radius is the square root of the sum of the squares of the blur radii that were actually applied. 


So in your case, that would be equivalent to blurring the image with a single Gaussian with $\hat{\sigma}$:
$$
\hat{\sigma} = \sqrt{8\sigma^2} = 2\sqrt{2} \sigma
$$