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No, this is impossible. It is always the case that $\chi(H) \leq |E|$ (or at least $\chi(H) \leq 2|E|$ if $E$ is finite). To see this, note that it's easy to define an injective partial function $c_0$ from a subset of $V$ to $|E|$ (or $2|E|$ if $E$ is finite) such that, for all $e \in E$, we have $|\mathrm{dom}(c_0) \cap e| \geq 2$ (simply enumerate the edges in order type $|E|$ and take care of them one at a time). Then any extension of $c_0$ to a function $c:V \rightarrow |E|$ will be a hypergraph coloring of $H$.