  Any finite word $w$ of length $n$ contains at most $n+1$ distinct palindromic
factors. If the bound $n+1$ is reached, the word $w$ is called rich. The number
of rich words of length $n$ over an alphabet of cardinality $q$ is denoted
$R_n(q)$. For binary alphabet, Rubinchik and Shur deduced that ${R_n(2)}\leq c
1.605^n $ for some constant $c$. We prove that $\lim\limits_{n\rightarrow
\infty }\sqrt[n]{R_n(q)}=1$ for any $q$, i.e. $R_n(q)$ has a subexponential
growth on any alphabet.
