jmc

Consider the polynomial $p(x)-5$. It has five zeroes, namely at the five points where $p(x)=5$. It follows that $p(x)$ must at least be a quintic polynomial. With a bit of experimentation, we see that it is indeed possible for $p$ to be a quintic. For example, the polynomial $p(x)=(x-2)(x-1)x(x+1)(x+2)+5$ satisfies these conditions: Hence, the least possible degree of $p$ is $\boxed{5}$.