Every body knows that if $V$ is the real vector space $\mathbb R[X]_{\leq n}$ of polynomials of degree at most $n$, then the linear map $$D:p\in V\mapsto p'\in V$$ given by derivation is nilpotent. It is indecomposable, in the sense that its Jordan canonical form has exactly one block.
The same thing happens with the maps $p\in V\mapsto p(X)-p(X-1)\in V$ given by “finite differences”, and the map $p\in V\mapsto \frac{p(qX)-p(X)}{(q-1)X}\in V$ (where $q$ is a scalar different form $1$) given by “quantum differentiation”, and one can come up with several other naturally occurring indecomposable nilpotent endomorphisms.
Are there natural examples of endomorphisms of finite dimensional vector spaces which are nilpotent and decomposable?
By natural I mean that it occurs in nature — one can of course construct artificial examples! Also the examples should be presentable to people learning linear algebra, so multiplicative extraordinary cohomology theories are not going to work ;-)
One natural thing to do is to look at the adjacency matrix of an acyclic directed graph, which is generally decomposable.
I don't know if you consider this as a cheat, but you can consider powers of indecomposable nilpotent operators (i.e "second derivative", "third derivative", etc). If $T$ is an indecomposable nilpotent operator that acts on an $n$-dimensional space then $T^i$ will be a decomposable nilpotent operator that decomposes as a direct sum of $i$ indecomposable nilpotent operators (on subspaces that has dimensions which can be calculated explicitly). This won't give you all possible examples but will give you enough to convince students that there are many kinds of nilpotent operators and it might be interesting to classify them.