Which is asymptotically larger: $\lg(\lg^* n)$ or $\lg^* (\lg n)$?

$\lg^* n$ is the number of times the logarithm function must be iteratively applied to $n$ before the result is less than or equal to 1.

Let us assume $\lg^* n = x$.

So, $\lg(\lg^* n) = \lg x$

And, $\lg^*(\lg n) = x - 1$ as we are applying logarithm once more thus reducing number of required iterations by 1.

Now, asymptotically $x - 1 > \lg x$, i.e. $\lg^* (\lg n)$ is asymptotically larger than $\lg(\lg^* n)$.