Applying DFT twice does not actually reverse an array. Instead, the first element stays in place while the rest of the array is reversed. Why?
The sequence is exactly what you should expect:
$$x[-n]=x[N-n]\tag{1}$$
Clearly, for $n=0$ $x[n]$ and $x[-n]$ have the same value.
It seems like you were expecting to see the sequence $x[N-1-n]$ instead of $x[N-n]$. If you wanted that sequence you would need to modulate the result of the first DFT:

N = 16;
x = 1:N;
c = exp(-1i*2*pi/N*(0:N-1));
x2 = real( fft( ( fft(x) .* c ) ) ) / N;    % real() just to remove rounding errors
x2 =

  16  15  14  13  12  11  10  9  8  7  6  5  4  3  2  1