Is there any example showing that the composition of morphisms is not necessarily associative?
I don't know how much category theory you know, but I guess you can look up any terms you don't recognize.
In a cateogry, composition is associative per definition. However, when generalizing categories to higher categories ($n$-categories), it is sometimes useful not to demand associativity, but only "weak associativity". Weak associativity is the same as associativity up to isomorphism in the layer above. That is, for any triple $f,g,h$ of morphisms where $gf$ and $hg$ are defined, there is an "isomorphism of morphisms" $F: (hg)f \rightarrow h(gf)$.
For example, if we define a path in a topological space $X$ to be a continuous function $\alpha:[0,1]\rightarrow X$, there is a well defined operation of "composition" of paths which is weakly associative, but not associative. In this case we get a 2-category where composition is associative up to homotopy.