How many distinct colors are needed to lower-bound the choosability of a graph?
Daniel Král and Jiří Sgall answered your question to the negative. From the abstract of their paper:


  A graph $G$ is said to be $(k,\ell)$-choosable if its vertices can be colored from any lists $L(v)$ with $|L(v)| \ge k$, for all $v\in V(G)$, and with $|\bigcup_{v\in V(G)} L(v)| \le \ell$. For each $3 \le k \le \ell$,
  we construct a graph $G$ that is $(k,\ell)$-choosable but not $(k,\ell+1)$-choosable.


So, $N(k)$ does not exist if $k\ge 3$. Král and Sgall also show that $N(2)=4$. Of course, $N(1)=1$.

Daniel Král, Jiří Sgall: Coloring graphs from lists with bounded size of their union. Journal of Graph Theory 49(3): 177-186 (2005)