Hardness of additive approximation to Graph Coloring problem.
Let $G$, with $\chi(G) \geq 3$ (otherwise you have an exact algorithm).

Now construct a graph $H$ consisting of $k$ copies $G_1,\ldots,G_k$ of $G$, with all possible edges between $u$ and $v$ for $u \in G_i$, $v \in G_j$, $i\not=j$.

Now use the additive constant algorithm for the graph $H$, which will find a coloring of, at most, $k\cdot\chi(G) + k$ colors. Since the colors in each copy of $G$ will be distinct, by the Pigeonhole principle, you have one of the copies with at most $\chi(G)+1$ distinct colors, which is better than a 2-approximation.