This question occurred to me while thinking on another one here, http://mathoverflow.net/q/219582/41291

Can one define *in an invariant way* a binary operation on vector spaces - let us denote it somehow suggestively by $(V,W)\mapsto V^{\otimes W}$ - with the property$$\dim(V^{\otimes W})=\dim(V)^{\dim(W)}?$$To avoid some complications, let us restrict to the case when both $V$ and $W$ are finite-dimensional.

Added later (and I should do it from the beginning as it is important): the construction from the linked question suggests that this operation seemingly should act on linear operators by assigning to $f:V\to X$ and $g:Y\to W$  certain operator (explicitly given there in terms of chosen bases)
$$
f\dagger g:V^{\otimes W}\otimes Y\to X\otimes W,
$$
with $\operatorname{rank}(f\dagger g)=\operatorname{rank}(f)\operatorname{rank}(g)$.

I'm aware that this looks even more impossible, but anyway - with bases it is done in that question.