I'm sure you're aware that linear recurrent sequences are well understood and can be solved exactly by a "closed formula". Once you enter the world of non-linear recurrences, even when restricting to constant coefficients, you can get all sorts of strange behavior even with some very simple looking recurrences. One example is the logistic map:
$$x_{n+1}=rx_n(1-x_n)$$
which is only exactly solvable for a few special values of the constant $r$. For the complex and chaotic behavior that happens with other values of $r$ you can check the wiki article, which has a nice bifurcation diagram and summary of the qualitative properties of this map.