Gluing filtered object from associated graded pieces So, I believe the following result is correct but do not know the exact reference (and not sure to what extent what I'm saying is true). If anyone could give a reference for this it would be great.
1) Consider three objects in some abelian category, $F_1, F_2, F_3$. Then objects with filtrations $F^1 \subset F^2 \subset F^3$ such that $F^1 = F_1, F^2/F^1 = F_2, F^3/F^2 = F_3$ are classified by triples $(\alpha, \beta, \gamma)$ of extension elements, $\alpha \in Ext^1(F_2, F_1), \beta \in Ext^1(F_3, F_2), \gamma \in Ext^1(F_3, F_1)$, such that the Yoneda product $\alpha \beta \in Ext^2(F_3, F_1)$ vanishes.
2) Similar result should hold for any amount of objects at least for the case of linear category in characteristic zero - the ways of gluing objects $F_1, ..., F_n$ into a filtered object $F^1 \subset F^2 \subset ... \subset F^n$ such that $F^k / F^{k-1} = F_k$ should be classified by Maurer-Cartan elements in the algebra $\bigoplus_{j>i}RHom^{\bullet}(F_j, F_i)$ (considered as either dgla or L-infinity algebra).
I also know that this type of questions frequently appear in the theory of mixed hodge structures (but unable to find any direct reference, too).
Edit: added forgotten $\gamma$
 A: Here is how I would think about this:  $\alpha \in Ext^1(F_2,F_1)$ corresponds to a short exact sequence $$0 \rightarrow F_1 \xrightarrow{i} F(1,2) \xrightarrow{p} F_2 \rightarrow 0.$$ 
Similarly $\beta \in Ext^1(F_3,F_2)$ corresponds to a short exact sequence $$0 \rightarrow F_2 \rightarrow F(2,3) \rightarrow F_3 \rightarrow 0.$$ 
The first short exact sequence induces a long exact sequence including
$$ Hom(F_3,F_2) \xrightarrow{\alpha \circ} Ext^1(F_3,F_1) \xrightarrow{i_*}  Ext^1(F_3,F(1,2)) \xrightarrow{p_*} Ext^1(F_3,F_2) \xrightarrow{\alpha \circ} Ext^2(F_3,F_1),$$ 
and from this one sees that an $F(1,2,3) \in Ext^1(F_3,F(1,2))$ exists such that $F(1,2,3)/F_1 = F(2,3)$ if and only if $\alpha \circ \beta = 0 \in Ext^2(F_3,F_1)$.  Furthermore, choices correspond to the image of $i_*$.  Perhaps this is the classification you desire.  I don't know of a reference, but the argument is just using basic triangulated category/homological algebra techniques.
Understanding filtered objects with 4 or more composition factors leads one quickly to Massey products.  Ambiguities tend to get out of hand unless one has something special going on in the case in hand.
