  For every semilattice $\mathcal{S}=(S,+)$, the set
$\mathrm{End}(\mathcal{S})$ of its endomorphisms forms a semiring under
point-wise addition and composition. We prove that the semiring of all
endomorphisms of the 3-element chain has no finite identity basis. This,
combined with earlier results by Dolinka (The finite basis problem for
endomorphism semirings of finite semilattices with zero, Algebra Universalis
61, 441-448 (2009)), gives a complete solution to the finite basis problem for
semirings of the form $\mathrm{End}(\mathcal{S})$ where $\mathcal{S}$ is a
finite chain.
