Long Division

One way to approach this problem is to notice that if:

$1 + \frac {1}{1 + \frac {1}{1 + \frac{1}{1 + ...}}} = x$

Then:

$1 + \frac{1}{x} = x$

Multiplying by x and shifting to produce a quadratic equation:

$x^2 - x - 1 = 0$

Of course this equation has two roots at:

$x = \frac{1 \pm \sqrt{5}}{2}$

But inspection of the original problem informs us that the quantity must be positive so that:

$x = \frac{1 + \sqrt{5}}{2}$