Let $R_q(n)$ denote the number of rich words of length $n$ over a fixed alphabet of size $q\geq2$. It is known that $R_q(n)^{1/n}\to1$. We establish a nonrecursive subexponential upper bound on the number of rich words, with its asymptotic growth quantified in terms of an iterated logarithm. For every $λ\in(1/2,1)$ and $γ>1$, there exists $c>0$ such that, for every positive function $φ$ satisfying $φ(n)\to\infty$ and $n/φ(n)\to\infty$, the functions \[ f(n)=\left[c\ln^{*}\!\left(\frac{n}{φ(n)}\ln q\right)\right]^{1/γ}, \qquad B(n)=q^{\frac{n}{φ(n)}+\frac{n}{(2λ)^{f(n)-1}}} \] satisfy $R_q(n)\leq B(n)$ for all sufficiently large $n$ and $B(n)^{1/n}\to1$. Here $\ln^{*}$ is the iterated natural logarithm, with value zero on $(0,1]$. The proof iterates a counting inequality based on palindromic factorization and controls the resulting multiplicative constants using the iterated logarithm.

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