If you set up the equations slightly differently it's easier to see:

x = 1 + 1/(1+x)

x*(1+x) = (1+x)+1

x^2+x = x +2

x^2=2

The core of the solution is recognizing that it can be reduced to a pair of algebraic equations rather than finishing off the computations. I was referring to the former in saying "could see how to answer it immediately." An extremely gifted child might also be able to solve the equations without pencil and paper, but that's a separate issue from abstract pattern recognition.

I solved both of them, slowly, in a sleep-deprived state. For the continued fraction, I first tried doing successive approximations to see what the answer "should" be... when I got 1.41 I figured that it was probably the square root of 2. So the next thing I did was to try squaring the expression, which wasn't exactly helpful, but it did lead me to notice that the continued fraction contained itself so I could use the algebra trick that Luke_A_Somers used.

I tried for maybe thirty seconds to solve it, but couldn't see anything obvious, so I decided to just truncate the fraction to see if it was close to anything I knew. From that it was clear the answer was root 2, but I still couldn't see how to solve it. Once I got into work though I had another look, and then (maybe because I knew what the answer was and could see that it was simple algebraically) I was able to come up with the above solution.

I spent around twenty seconds looking at it and gave up. Then I came back fifteen minutes later, spent an additional twenty seconds looking at it and figured it out. I'm not sure what that says about my intelligence/pattern-recognition skills, but it probably says bad things about my conscientiousness.