Let us look at the graph $\Gamma$ whose vertices $V$ consists of the elements $v$ in the lattice $\mathbb{Z}^2$, and whose edges consist pairs of elements in $v_1,v_2 \in \mathbb{Z}^2$ such that $|v_1-v_2|=1$. In $\Gamma$ we may look at different families of paths, where by path I mean a sequence of oriented edges $\{e_n\}$ in $\Gamma$ such that the end-point of $e_k$ is the starting point of $e_{k+1}$.
One such family could be all paths starting at $0$ and never backtracking. It is easily seen that the number of such paths of length $n$ will be given by $4\cdot 3^{n-1}$ (you have four possible direction to choose from in the first step and three in all subsequent steps). Another family would be paths consisting only of edges going either up or to the right (in the case the number of paths will be $2^n$).
A more interesting family would the set of paths which are self-avoiding, i.e. they never intersect themselves. It is a well-known conjecture that the number of selfavoiding paths of length $n$ - called $c_n$ here - are given asymptotically be the formula $$ c_n = c^n n^{11/32} + o(c^nn^{11/32})$$ where $c$ is a suitable real number (the socalled connective constant).
I would be interested in other natural families of curves (preferable more well-understood or easier to analyze than the self-avoiding ones) which also satisfy that the number of curves of length $n$ in such a family - called $a_n$ here - would satisfy $$ a_n = a^n p(n^{\alpha}) + o(a^np(n^{\alpha}))$$ where $a$ is a suitable real number, $\alpha$ is a non-zero rational number and $p$ is a polynomial of degree at least 1.