@SimpleStacker gave the right answer.

If you don't see how they got the quadratic equation, notice that this is a infinite continued fraction. So, everything below the first numerator (i.e. the infinitely long denominator) is equal to $x$ itself. Hence, $x = 1+1/x$.

To solve the quadratic equation, use the usual $\frac{-b\pm\sqrt{b^2-4ac}}{2a}$ formula, and discard the negative solution as explained in another comment. We then get $\frac{1+\sqrt{5}}{2}\simeq 1.618$ which is the golden ratio.