I know it's not the question, but, anent now-user76479's / I guess-then Arul's [comment](https://mathoverflow.net/questions/219630/exponentiation-of-vector-spaces#comment541600_219630) and your [response](https://mathoverflow.net/questions/219630/exponentiation-of-vector-spaces#comment541602_219630), one way to make use of an algebra structure on $W$, if it is present, is to define $V^{\otimes W} = V^{\otimes\operatorname{Hom}_\text{alg}(W, k)}$ (i.e., one ‘factor’ in the tensor product for each element of the set of algebra homomorphisms), where $k$ is the underlying field.  Of course this can fail to exhibit the desired dimension behaviour for a randomly chosen finite-dimensional $k$-algebra $W$, but, if $W$ is an [étale algebra](https://en.wikipedia.org/wiki/%C3%89tale_algebra), then it works as desired.