problem: If $x = 1 + \frac{x}{1 + \frac{x}{1+ \frac{x}{1 + \cdots}}}$, then what is $x$?
solution: Notice that the in the equation $$x = 1 + \frac{x}{\left(1 + \frac{x}{1+ \frac{x}{1 + \cdots}}\right)},$$the term in the parentheses is the same as the definition of $x$. It thus follows that $$x = 1 + \frac{x}{x} = \boxed{2}.$$