Every context-free language has either polynomial growth or exponential 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:  
> Roberto Incitti:  
> "The growth function of context-free languages"  
> Theoretical Computer Science 255 (2001), Pages 601-605 

> Martin R. Bridson, Robert H. Gilman:  
> "Context-Free Languages of Sub-exponential Growth"  
> Journal of Computer and System Sciences 64 (2002), Pages 308-310

And for a given context-free grammar, one can decide in polynomial time whether the generated language has polynomial or exponential growth:  
> Pawel Gawrychowski, Dalia Krieger, Narad Rampersad, Jeffrey Shallit:  
> "Finding the Growth Rate of a Regular or Context-Free Language in Polynomial Time.   
> International Journal of Foundations of Computer Science 21 (2010), Pages 597-618