Combine lvs without direct sum or tensor product? For a finite number of finite dimension linear vector spaces we have two common, well defined means of combining them to make a new vector space:  Direct sum/direct product and tensor product.  What other well-defined methods exist, if any? 
 A: Another important basic one is that given two vector spaces $V, W$ we can produce the vector space $[V, W]$ (also notated $\text{Hom}(V, W)$) of linear maps $V \to W$. More complicated examples can be built by iterating these constructions to form "polynomials" such as $(V \otimes W) \oplus (V^{\otimes 2} \otimes W^{\otimes 2})$. 
Conversely one might ask for a classification of all such things. (This part of the answer will require much more background to understand.) This requires a way to make more precise what we really mean by a way of combining vector spaces to form a new vector space. One possible formalization is that we want functors: all of the examples discussed so far are functors, although $[V, W]$ is contravariant in $V$. So one might ask for a classification of functors
$$\text{Vect} \times \text{Vect} \to \text{Vect}$$
or something like that, maybe finite-dimensional vector spaces specifically. A very large class of examples, which I think exhausts all "reasonable" examples, is given by the following class of "multivariate power series": let $C_{n, m}$ be any family of representations of the product of symmetric groups $S_n \times S_m$, and take the functor
$$(V, W) \mapsto \sum_{n, m \ge 0} C_{n, m} \otimes_{S_n \times S_m} V^{\otimes n} \otimes W^{\otimes m}$$
where, if $V, W$ are two representations of a group $G$, $V \otimes_G W$ is the quotient of $V \otimes W$ by the relation $gv \otimes w \sim v \otimes gw$ for all $g \in G$. The single-variable version of this construction is called taking Schur functors, so one might call these multivariate Schur functors. A similar construction works for more than two inputs. 
This describes constructions taking in vector spaces and producing vector spaces; I actually don't know the answer to the corresponding question for finite-dimensional vector spaces. We can restrict our attention to "polynomials" with f.d. coefficients, but this doesn't exhaust all examples; for example, in one variable the "power series"
$$V \mapsto \sum_{n \ge 0} \Lambda^n(V)$$
where $\Lambda^n(V)$ is the $\text{sign} \otimes_{S_n} V^{\otimes n}$ (so this whole construction is the exterior algebra) sends f.d. vector spaces to f.d. vector spaces despite not being a "polynomial." I actually don't know a complete classification here. 
