Let $C$ and $L$ be plane algebraic curves of degree $n$ and $1$ defined over the real numbers. For $k=1,\ldots n$, let $h(k)$ be the number of real points where $L$ and $C$ meet with multiplicity $k$ and let $j(k)$ be the number of non-real points where $L$ and $C$ meet with multiplicity $k$. Thus $j(k)$ is even for all $k$, and $\Sigma h(k)+\Sigma j(k)=n$.
Let $S_n$ be the cardinality of the set of all pairs $(h,j)$ where $h$ and $j$ are functions from the positive integers to the non-negative integers that satisfy these conditions. Then $S_n$ measures the number of different ways a real plane curve of degree $n$ can intersect a real line, where the definition of ``different'' is implicit in the above.
It's easy to write down a generating function for $S_n$, and then to compute the first several values (beginning with $1,3,4,9,12$) and then to look up this sequence on the OEIS to confirm its accuracy.
After computing a great many more values, it appears that $S_n$ behaves very much like $C\cdot exp(n^\alpha)$ for some constants $C$ and $\alpha$. (In fact, $\alpha$ is in the vicinity of $.64$.)
Is there some simple reason I should have expected this?