, 2013

Solution:

$\boxed{\dfrac{2013}{2014}}$

Let $f(n)$ denote the corresponding expression with the digit $2$ appearing exactly $n$ times. Then $f(1) = \dfrac{1}{2}$ and for $n > 1$, $f(n) = \dfrac{1}{2 - f(n-1)}$.

By induction using the identity
$$\frac{1}{2 - \frac{N-1}{N}} = \frac{N}{N+1},$$
we have $f(n) = \dfrac{n}{n+1}$ for all $n \geq 1$, so $f(2013) = \dfrac{2013}{2014}$.