> It’s very subjective, but I think the uniform stsrts looking reasonably good at a sample size of 8. The exponential however takes much longer to converge to a normal.

That's a good observation. The main idea behind the Central Limit Theorem is to take the Fourier Transform, operate and then go back. After that, after normalization the result is that the new distribution for the sum of N variables is something like

  Normal(X) + 1/N * "Skewness" * Something(X) + 1/N^2 * IDont * Remember(X) + ...

The uniform distribution is symmetric, so skewness=0 and the correction decrease like 1/N^2.

The exponential distribution is very asymmetrical and and skewness!=0, so the main correction is like 1/N and takes longer to dissapear.