jmc

Consider the polynomial $q(x)=p(x)-x.$ This polynomial becomes 0 at $x=1,$ 2, 3, 4, 5, and 6, so it has $x-1,$ $x-2,$ $x-3,$ $x-4,$ $x-5,$ and $x-6$ as factors. Also, $p(x)$ is a monic polynomial of degree 6, so $q(x)$ is a monic polynomial of degree 6. Hence, $$q(x)=(x-1)(x-2)(x-3)(x-4)(x-5)(x-6).$$Then $q(7)=6\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1=720,$ so $p(7)=q(7)+7=\boxed{727}.$