  We prove a transcendence theorem concerning values of holomorphic maps from a
disk to a quasi-projective variety over $\overline{\mathbf{Q}}$ that are
integral curves of some algebraic vector field (defined over
$\overline{\mathbf{Q}}$). These maps are required to satisfy some integrality
property, besides a growth condition and a strong form of Zariski-density that
are natural for integral curves of algebraic vector fields. This result
generalizes a theorem of Nesterenko concerning algebraic independence of values
of the Eisenstein series $E_2,E_4,E_6$. The main technical improvement in our
approach is the replacement of a rather restrictive hypothesis of polynomial
growth on Taylor coefficients by a geometric notion of moderate growth
formulated in terms of Value Distribution Theory.
