  Given a complex projective algebraic variety $X$ we define $ h(X)$ as the
largest $n$ such that the $n$-th symmetric power of $X$ is (Brody) hyperbolic.
Using Nevanlinna theory for algebroid maps, we give non-trivial lower bounds
for $ h(X)$. From an arithmetic point of view, the problem is closely related
to the finiteness of algebraic points of bounded degree in varieties over
number fields. We provide explicit applications of our results in the case of
curves embedded in surfaces and in the case of subvarieties of abelian
varieties.
