One does not have to go all the way to 7-th degree polynomials to see this behaviour. Consider $f(x) = x^3-5x$. If you start the Newton-Raphson iteration from initial guess $1$, you would simply oscillate between $1$ and $-1$. This is an unstable equilibrium, but clearly one that you might actual...

As the other answers already noted, there are cases where the Newton iteration does not converge. One interesting question is for how "many" values this can occur and whether it's a set or full measure or a Null set. As it turns out, there are indeed polynomials for which the Newton process does ...