Example 1. (1987 Math Summer Camp) Solve the equation
$$
x=\sqrt{1+\sqrt{1+\sqrt{1+x}}}
$$
Let $1+x=u$, then
$x=\sqrt{1+\sqrt{1}+\sqrt{u}}$ which means $\sqrt{u}=\sqrt{1+x}$
$=\sqrt{1+\sqrt{1+\sqrt{u}}}$, the structure of the equation is similar to the original, so
$x=\sqrt{u}$, that is $x=\sqrt{1+x}$, solving this gives $x=\frac{1+\sqrt{5}}{2}$.