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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?

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    You can also say that $M$ is parallelizable. This is true for any dimension as long as the tangent bundle is trivial. – 2017-01-07
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    Right, I didn't list that one because I was actually thinking it was almost part of the definition of trivial tangent bundle. – 2017-01-07

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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.)