Lets say we have a manifold $\mathcal{M}$ and the frame bundle $F\mathcal{M}$ has as structure group some subgroup $G$ of $GL(n,\mathbb{R})$. In such a case one can find the "invariant tensor(s)" i.e. tensors that transform according to the trivial representation of $G$.
1) Could somebody give me some examples of this? Like a list with three entries: the manifold $\mathcal{M}$, the group $G$ and the invariant tensor(s).
2) In particular if $G$ is trivial (this is the same than saying $\mathcal{M}$ is parallelizable) all tensors are invariant right? How is it possible to have tensors that do not change components at all?
I think that there is a partial misunderstanding in your question. You say that $F\mathcal M$ "has as structure group some subgroup $G$ of $GL(n,\mathbb R)$". The frame bundle itself always has structure group $GL(n,\mathbb R)$, but it may be possible to choose a reduction to a certain subgroup $G\subset GL(n,\mathbb R)$. But this is a choice, which introduces an additional structure on $\mathcal M$, and the invariant tensors depend on this additional structure.
For some groups $G$, such a reduction can be chosen on any manifold. For example putting $G=O(n)$ such a reduction is equivalent to the choice of a Riemannian metric $g$ on $\mathcal M$. The simplest examples of invariant tensors are then the metric $g$ itself, its inverse or the volume density determined by $g$.
For other groups $G$, reductions exist only under additional assumptions on $\mathcal M$. This is the case for the trivial group $\{I\}\subset GL(n,\mathbb R)$, for which a reduction exists only for parallelizable manifolds. But again the reduction itself is a choice, namely the choice of a global trivialization of the tangent bundle. (And there are many such choices, fixing one they are described by all smooth functions $\mathcal M\to GL(n,\mathbb R)$. Fixing such a global trivialization, you get global frames for all tensor bundles, and the invariant tensors then are exactly those tensors which have constant coordinates in that global frame.