Geometric structures possible given trivial tangent bundle of even dimension Let $M$ be a $2n$-dimensional manifold with trivial tangent bundle
$$
TM \cong M \times \mathbb R^{2n}.
$$
I know that $M$ admits the following (using the structure group or other methods):
(1) An orientation 
(2) A volume form
(3) A riemannian metric
(4) An almost complex structure
(5) And also these
Can any other structures be put on $M$ or is this a complete list? Does the fact that TM is actually a product imply any form of integrability?
 A: It's certainly not a complete list. 
How many such structures there are depends on exactly what you mean by "structures on $M$." One reasonable way to interpret that phrase would be $G$-structures on $M$ (i.e., reductions of the tangent bundle to some subgroup $G\subseteq GL(2n,\mathbb R)$). All of the structures you mentioned are of this type -- for example, an orientation is a $GL(2n,\mathbb R)^+$-structure; a volume form is an $SL(2n,\mathbb R)$-structure; a Riemannian metric is an $O(2n)$-structure; etc.
A trivialization of the tangent bundle of $M$ represents a reduction of the structure group to the trivial group, so for any subgroup $G\subseteq  GL(2n,\mathbb R)$ whatsoever, you can put a $G$-structure on $M$. 
Typically, there will be uncountably many distinct such structures. For example, in dimension $4$, let $a\in (0,1)$ and define a subgroup $G_a\subseteq  GL(4,\mathbb R)$ by
$$
G_a = \left\{ \left(
\begin{matrix} 
t & 0 & x & 0\\
0 & t^{-a} & y & 0\\
0 & 0 & 0 & 0\\
0 & 0 & 0 & 0
\end{matrix}
\right):
t>0, \ x,y\in\mathbb R\right\}.
$$
These three-dimensional groups all have Lie algebras of Bianchi type VI (see also this paper), so they are nonisomorphic for different choices of $a$. Thus no two of these $G$-structures are isomorphic to each other.
(This construction works in any dimension bigger than $2$, but I used dimension $4$ because you stipulated that you want the manifold to have even dimension. Off the top of my head, I don't know exactly what happens in dimension $2$ -- I suppose it's possible that there are only finitely many nonisomorphic $G$-structures in that case.)
