  The $q$-color Ramsey number of a $k$-uniform hypergraph $G,$ denoted
$r(G;q)$, is the minimum integer $N$ such that any coloring of the edges of the
complete $k$-uniform hypergraph on $N$ vertices contains a monochromatic copy
of $G$. The study of these numbers is one of the most central topics in
combinatorics. One natural question, which for triangles goes back to the work
of Schur in 1916, is to determine the behaviour of $r(G;q)$ for fixed $G$ and
$q$ tending to infinity. In this paper we study this problem for $3$-uniform
hypergraphs and determine the tower height of $r(G;q)$ as a function of $q$.
More precisely, given a hypergraph $G$, we determine when $r(G; q)$ behaves
polynomially, exponentially or double-exponentially in $q$. This answers a
question of Axenovich, Gy\'{a}rf\'{a}s, Liu and Mubayi.
