The ring generated by all functions from a set to itself Let $S$ be a finite set.  Now $\mathop{End}(S)$ is a monoid, and we may build a ring $R$ by allowing formal sums of functions.
Preliminary questions, since $R$ is surely well-known:  What is it called?  In a general category, what is the name of the construction that builds a ring out of the endomorphisms of an object?
My real question is about $A = R \otimes \mathbb{C}$.  Restricting to the subalgebra generated by the automorphisms of $S$, it is clear that understanding the representation theory of symmetric groups is a prerequisite for understanding $A$.  Are the symmetric groups on sets of size smaller than $| S |$ responsible for all nontrivial properties of $A$?  If so, what suitably strong formulation of inclusion/exclusion is at work?
 A: The monoid of all maps on $n$ letters is denoted $T_n$ and called the full transformation monoid.  Your intuition is both right and wrong.  The irreducible representations of $T_n$ are in bijection with irreducible representations of all symmetric groups of degree at most $n$.  The character table is block upper triangular with diagonal blocks character tables of symmetric groups of degree at most $n$.  Thus inverting the table, which is how you decompose characters into irreducibles, is a sort of generalized inclusion-exclusion.  Putcha computed the character table, although Hewitt and Zuckermann stated a portion of the result without a complete proof in 1957.  But this is only the semisimple part of the story.
The algebra of $T_n$ is not semisimple.  For example, nobody knows what the projective indecomposables look like in general.  The algebra is quasihereditary and so has finite global dimension.  Putcha computed a lot of information about the quiver.  Ponizovsky showed $T_1,T_2,T_3$ has finite representation type, Ringel showed $T_4$ has finite representation type and Putcha showed the representation type is infinite in all other cases.  I think this part of the representation theory is not completely controlled by symmetric groups.
Update 1/15.
In case you are still interested I just put up a paper 
http://arxiv.org/abs/1502.00959 computing the global dimension of this algebra.  From the paper you can get a very precise idea of how the representations of the symmetric groups of different degrees interact to control the Ext spaces between simple modules in a non trivial way. Also the paper gives a more accurate survey of the representation theory of this algebra than my answer.
