Every context-free language has either polynomial growth or polynomial growth: In the notation of the question poser:
- Either there is a polynomial $p$ so that $w_n\le p(n)$ for all $n$
- Or there exists a $c>1$, so that $w_n\ge c^n$ for infinitely many $n$.
This has been shown for instance in:
Martin R. Bridson, Robert H. Gilman:
"Context-Free Languages of Sub-exponential Growth"
Journal of Computer and System Sciences 64, (2002), Pages 308-310