This question occurred to me while thinking on another one here, Name for an operation on matrices?

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) $$ V^{\otimes W}\otimes Y\to X\otimes W. $$ I'm aware that this looks even more impossible, but anyway - with bases it is done in that question.