Physicists computing multiloop Feynman diagrams have introduced various
techniques and conjectures that involve the notion of Degree of Transcendentality (DoT). From what I understand one defines


1) $DoT(r)=0$, r rational 


2) $DoT(\pi^k)=k$, $k \in {\mathbb N}$,


3) $DoT(\zeta(k))=k$,


4) $DoT( a \cdot b)= DoT(a)+DoT(b)$


One then proves for example that the $\ell$-loop contribution to a certain scaling function in $N=4$ Supersymmetric gauge theory consists of a sum of terms all of which have DoT equal to $2 \ell-2$. 


This can't be rigorous mathematically, since it is not even known that $\zeta(2n+1)$ is transcendental, but is there some circle of ideas, or conjecture in mathematics that if true would give a precise definition to DoT?



Degree of Transcendentality and Feynman Diagrams
Your *transcendentality* reminds me about the Institute of Algebraic Meditation at Höör (Sweden). To be honest, your definition corresponds to what is known as the *weight* of a (multiple) zeta value (see Michael Hoffman's <http://www.usna.edu/Users/math/meh/mult.html>, especially the [references on MZVs](http://www.usna.edu/Users/math/meh/biblio.html)). These indeed occur in the computation of Feynman's diagrams. As for conjectures related to the transcendental number theory tag, a belief is that $\pi$ and odd zeta values $\zeta(3)$, $\zeta(5)$, etc, are algebraically independent over the rationals.


