jmc

We build a polynomial $P(x)$ by starting with the equation $x=\sqrt{1+\sqrt{6}}$ and trying to generate an equation for $x$ with only rational coefficients. To start, square this equation, giving $$x^2=1+\sqrt{6}.$$If we subtract $1$ and then square again, we see that $$(x^2-1{)}^2=(\sqrt{6}{)}^2$$or $x^4-2x^2+1=6.$ Thus, $x^4-2x^2-5=0,$ so we have shown that $\sqrt{1+\sqrt{6}}$ is a root of $x^4-2x^2-5.$ Therefore, we have $P(x)=x^4-2x^2-5,$ and so $P(1)=1-2+5=\boxed{-6}.$