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

Here is how $$\lg^\ast n$$ is defined mathematically:

$\lg^\ast n = \min \{i \geq 0 : lg^{(i)} n \leq 1\}$

Which practically means, $$\lg^* n$$ is the number of times the logarithm (base $$2$$) function can 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:

\begin{aligned} x - 1 &> \lg x \\ \lg^\ast (\lg n) &> \lg(\lg^\ast n) \end{aligned}