For a finitely generated extension of fields $K/k$, let us define "$S\_{K/k}$" to be the minimum of the degrees $[K:\ell]$ where $\ell/k$ ranges over the purely transcendental subextensions of $K$ with $\mathrm{tr.deg}(K/k) = \mathrm{tr.deg}(\ell/k)$.



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> Does there exist a tower $K\_{3}/K\_{2}/K\_{1}$ of finitely generated extensions such that $S\_{K\_{3}/K\_{1}} \ne S\_{K\_{3}/K\_{2}} \cdot S\_{K\_{2}/K\_{1}}$?
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Transitivity of an invariant of finitely generated field extensions
Yes, I think so. Let $P\in \mathbb{C}[t]$ be a polynomial of degree $\geq 3$, with no multiple root and $P(0)\neq 0$.
Take $K\_1=\mathbb{C}$, $K\_2=\mathbb{C}(t)[x]/(x^2-P(t))$, $K\_3=K\_2[u]/(u^2-t)$. Then $S\_{K\_2/K\_1}=2$, $S\_{K\_3/K\_2}=[K\_3:K\_{2}]=2$, but $S\_{K\_3/K\_1}=2$ because $K\_3=\mathbb{C}(u)[x]/(x^2-P(u^2))$. 


