Example of a preclosure that is not a closure

A topological example: let $X$ be a space and define $\phi(A) = \{x \in X: \exists (a_n) \in A^\mathbb{N}: a_n \to x\}$, the sequential closure operation.

For metric spaces this would just be the normal closure, but for the space $X$ the Arens' space, as defined here, we have that $\phi(\mathbb{N}\times \mathbb{N}) = (\mathbb{N} \times \mathbb{N}) \cup \mathbb{N}$ and $\phi(\phi(\mathbb{N} \times \mathbb{N})) = X \neq \phi(\mathbb{N} \times \mathbb{N})$, showing that $\phi$ does not obey 3.

Sequential closure is one of the motivating examples for studying the Cech closure spaces as a generalisation of topological spaces. There are also analysis examples in measure theory IIRC (convergence a.e. or convergence in measure or some such notion)

The answer is already in the comments so to post it here explicitly, set $X=\{1,2,3\}$, define $\varphi_3:\mathscr{P}(X)\rightarrow \mathscr{P}(X)$ as following: $\varphi_3:A\mapsto A\cup\{n-1\mid n\in A\setminus\{1\}\}$. That means: $$\emptyset\mapsto\emptyset,\{1,2,3\}\mapsto \{1,2,3\}$$ $$\{1\}\mapsto \{1\}, \{2\}\mapsto\{1,2\},\{3\}\mapsto\{2,3\}$$ $$\{1,2\}\mapsto \{1,2\}, \{1,3\}\mapsto \{1,2,3\}, \{2,3\}\mapsto\{1,2,3\}$$