(Much of this has basically been said by someone in the comments already.)

Here is a way of making examples of topologies on Set. Let $\mathcal C$ be a class of sets. Define a topology on Set by saying that a sieve $S$ on an object $X$ is a cover if and only if for every $Y\in\mathcal C$ every morphism $Y\to X$ belongs to $S$. An object is a point (in the sense that the only cover of that object is its maximal sieve) if it belongs to $\mathcal C$, and more generally if it is a retract of such an object. If we enlarge $\mathcal C$ to make it closed under retraction the topology is unchanged. Any presheaf that is a right Kan extension from (the full subcategory) $\mathcal C$ is a sheaf, and conversely any sheaf coincides with the Kan extension of its restriction.

(All of that is valid for any category, not just for Set.)

In the case when $\mathcal C$ consists of just a singleton, this topology on Set is the canonical one, where the sheaves are the representable funtors. If $\mathcal C$ has at least one nonempty set in it then the sheaf is subcanonical.

In general a class of sets closed under retraction must be either: all sets with cardinality less than a fixed cardinal, or all non-empty sets with cardinality less than a fixed cardinal. For example, $\mathcal C$ might be all nonempty sets having at most $n$ elements, or all nonempty finite sets, or all nonempty countable sets, or all nonempty sets (or any of these together with the empty set).

I suppose there are examples of topologies on Set not of this kind. That is, I suppose that there is a topology $T$ on Set such that $T$ does not coincide with the largest one that has the same points as $T$. Does somebody know?

EDIT: I realize now that of course there are perfectly everyday examples not of that kind. For example, the topology generated by finite covers in the ordinary sense. ($X$ is covered by ${Y_i\to X}$ if $X$ is the union of $Y_1,\dots ,Y_n$.)