What is the (transcendence) degree of the field $\mathbb{Q}^*$ generated by Kontsevich-Zagier periods over $\overline{\mathbb{Q}}$? What is its algebraic closure, $\overline{\mathbb{Q}^*}$?
What are the automorphism groups $Gal(\mathbb{Q}^*|\overline{\mathbb{Q}}$), $Gal(\overline{\mathbb{Q}^*}|\mathbb{Q}^*)$ and $Gal(\overline{\mathbb{Q}^*}|\overline{\mathbb{Q}})$?
How are these related to the motivic Galois group?
I was going to write this as a comment, but it turned out too long so I am posting it as an answer. The Kontsevich-Zagier periods include periods of motives over $\overline{\mathbb{Q}}$ (well at least, they include periods of cohomology groups of varieties): The periods of $H^n(X)$ of a smooth variety $X$ over $\overline{\mathbb{Q}}$ are the numbers that appear as the image of the pairing between algebraic de Rahm cohomology and singular homology, i.e. roughly speaking just the integrals of closed algebraic $n$-forms over closed singular $n$-chains. I will consider your question about periods of motives instead of your $\mathbb{Q}^\ast$ (in fact, it might be that every Kontsevich-Zagier period is a period of a motive too - you should check the literature).
For periods of motives, instead of asking for the transcendence degree of the full space of periods of motives we can ask about transcendence degree of the field $Per(M)$ generated by the periods of a given motive $M$ over $\overline{\mathbb{Q}}$. Grothendieck's period conjecture predicts that the transcendence degree of $Per(M)$ should be equal to the dimension of the motivic Galois group of M (the hypothesis that $M$ is over $\overline{\mathbb{Q}}$ or more generally a field of transcendence degree zero is crucial here). That the former is bounded from above by the latter follows from a standard argument (see for instance, Deligne's paper on absolute Hodge cycles in LNM 900). But the reverse inequality is a very deep statement and one of the most important conjectures in arithmetic geometry. I would suggest reading Ayoub's short and introductory article in the European Math. Soc. Newsletter (2014). The article might also answer your follow-up questions about motivic Galois groups.
