Only 4 Square Roots

Consider the following:

$\begin{aligned} y & = \sqrt{x+\sqrt{x+\sqrt{x+\sqrt{\color{#3D99F6}x+a}}}} & \small \color{#3D99F6} \text{Putting }x+a = a^2 \\ & = \sqrt{x+\sqrt{x+\sqrt{x+\sqrt{\color{#3D99F6}a^2}}}} \\ & = \sqrt{x+\sqrt{x+\sqrt{x+a}}} \\ & = \sqrt{x+\sqrt{x+a}} \\ & = \sqrt{x+a} \\ & = a \end{aligned}$

Implying that if $\sqrt{x+\sqrt{x+\sqrt{x+\sqrt{x+\color{#D61F06}a}}}} = \color{#D61F06}a$ , then $x+a=a^2$ $\implies x = a^2 - a$ . For $a=2018$ $\implies x = 2018^2 - 2018 = \boxed{4070306}$