Thank you to Achim Krause for pointing out that the first version was broken. Let's try again. 

Let $k$ be a finite field. For a set $X$, let $kX$ be the free vector space on $X$. Let $\bigwedge^{\bullet} kX$ be the exterior algebra on $kX$. Then $X \mapsto \bigwedge^{\bullet} kX$ becomes a functor in an obvious way.

Choose any scalars $a_0$, $a_1$, $a_2$, ... in $k$ and define the natural transformation of $\bigwedge^{\bullet} kX$ by multiplying $\bigwedge^{j} kX$ by $a_j$. This gives infinitely (even uncountably) many natural transformations from $X \mapsto \bigwedge^{\bullet} kX$ to itself as a functor from finite sets to finite sets (or even from finite sets to vector spaces).

When studying functors $F$ from the category of finite sets or related categories, one usually wants to impose some sort of finite generation condition, saying roughly that there is some integer $N$ such that any subfunctor of $F$ which agrees with $F$ on sets of size $\leq N$ is the same as $F$. One does this precisely to avoid this sort of trickery with the functor $X \mapsto 2^X$. For example, [Eric Ramos, Graham White and I][1] classify functors from FI to FinSet with a finite generation hypothesis and my student [John Wiltshire-Gordon][2] classified functors from FinSet to $\mathbb{Q}$-Vect under a similar hypothesis. These are the two papers I know which come closest to studying functors from FinSet to FinSet.


  [1]: https://arxiv.org/abs/1804.04238
  [2]: https://arxiv.org/abs/1406.0786