Continued fractions with $n$ layers Solve the equation 
$$x=2+\dfrac1{2+\dfrac1{...2+\dfrac1{2+\dfrac1x}}}$$
Where there are n layers in the fraction

the first thing to observe is that the number of layers doesn't matter. If $x = 2 + \frac{1}{x}$ it solves your equation, and simple continued fractions have a unique assigned value.
It's easy to solve $x = 2 + \frac{1}{x}$ though, just subtract two then multiply up to get $x^2-2x - 1 = 0$.
