2016 is coming!

$x^{2016} - 2016x - 2 = 0$

$x^{2016} = 2016x + 2 .... (1)$

Putting $x = x_{1}, x_{2}, ....., x_{2016}$ in $(1)$ repeatedly, and add them, we get

$x_{1}^{2016} + x_{2} ^{2016} + ........ + x_{2016}^{2016} = 2016(x_{1} + x_{2} + ..... + x_{2016}) + 4032$

Using Vieta, $x_{1} + x_{2} + .... + x_{2016} = 0$

So the value of expression is $x_{1}^{2016} + x_{2} ^{2016} + ........ + x_{2016}^{2016} = 4032$

$\begin{aligned} x^{2016}-2016x-2 & = 0 \\ x^{2016} & = 2016x+2 \\ \sum x^{2016} & = \sum \left(2016x+2\right) \\ & = 2016 \sum x+ \sum 2 \\ & = 2016\left(0\right)+2016\cdot 2 \\ & = \boxed{4032} \end{aligned}$

Some other solution... By Newton's sums

$x^{2016}_1+ x^{2016}_2+ \ldots + x^{2016}_{2016} = S_{2016}$

$S_{2016} + S_{2015} \cdot a_{1} + \ldots + S_1 \cdot a_{2015} +2016 \cdot a_{2016} = 0$

Since $a_1, a_2, a_3, ... a_{2014}=0$ and $S_1 = -a_1 = 0$ , then

$S_{2016} + S_{2015} \cdot 0 + \ldots + 0 \cdot a_{2015} +2016 \cdot a_{2016} = 0$

$S_{2016} + 2016 \cdot a_{2016} = 0$

$S_{2016} + 2016 \cdot (-2) = 0$

$S_{2016} = 4032$