2016 is absolutely awesome 14

Since $f(f(x, y), z) = f(x, f(y, z))$ , we have that: $\dfrac{1}{\dfrac{1}{x+y} + z}=\dfrac{1}{x +\dfrac{1}{y+z}}$ Simplifying, we have that

$x +\dfrac{1}{y+z}=\dfrac{1}{x+y} + z$

$x(y + z)(x + y) + (x + y) = (y + z) + z(x + y)(y + z)$

$x ((y + z)(x + y) + 1) = z ((x + y)(y + z) + 1)$

$(x - z) ((y + z)(x + y) + 1) = 0$

Since $x$ and $z$ are distinct, $x - z \ne 0$ so we may divide through by $x - z$ to obtain

$(y + z)(x + y) + 1 = 0$

$(y + z)(x + y) = -1$

Since $x + y + z = 2016$ , $y + z = 2016 - x$ and $x + y = 2016 - z$ so $(2016 - x)(2016 - z) = -1$

Since $x, y, z$ are integers, we have that $x = 2015$ and $z = 2017$ (or the other way around).

In either case, $y = 2016 - x - z = 2016 - 2015 - 2017 = \boxed{-2016}$ .