How to express the rest of division by three, with very elementary functions? Is it possible to express 
$\; "\!n\pmod 3\!"\;$ 
with combinations of the functions plus, minus, multiplication, division and exponentiation in $\mathbb C$ or preferably in $\mathbb Z[i]$?

I'ts not allowed with modulo or integer parts, and the inverses are only valid if they can be expressed as above. Also all sums, products etc must be finite.

There is a simular question Closed-Form Modular Arithmetic where trigonometric functions and logarithms are allowed.

If it isn't possible, how to prove that?
If complex exponentiation is OK, then you may do it. The map $f: n \mapsto \exp(2\pi n/3)$ takes ${\mathbb Z}$ to the three cubic roots of $1$:  $1$, $\gamma=\exp(2\pi i /3)$ and $\overline{\gamma}$. Now pick a polynomial $P$ that maps $1$ to $0$, $\gamma$ to $1$ and $\overline{\gamma}$ to $2$. Then $P\circ f$ will do the job. Same principle for mod $N$.

Later edit: In the general case set $\gamma=\exp(2\pi i/N)$ and define $e_N(z)=\frac{1}{N} \left(z^{N-1}+z^{N-2} + \cdots +z +1\right) $. Then  $e_N$ maps $1$ to $1$ but every $\gamma^k$, $1\leq k&lt;N$ to zero. The wanted function is then:
$$ h_N(n) = \sum_{k=1}^N k \ e_N \left( \ \gamma^{n-k}  \right), $$
which maps $n$ to $n\  (\mbox{mod}\; N)$. What is really happening here is that we are doing a forward + backward  Fourier transform of the modulus function.
$\frac{3t^2 - t}{2i\sqrt{3}}$ where $t=e^{2\pi in/3} + e^{-2\pi in/3}$