Wellpowered means that for every scheme $X$, the subobject lattice of monormophisms $Y \to X$ is essentially small; regularly wellpowered means that for every scheme $X$, the regular subobject lattice of regular monomorphisms $Y \to X$ (being a regular mono means that $Y \to X$ is the equalizer of some pair of maps) is essentially small. Wellpoweredness implies regular wellpoweredness, but not conversely.

Let me sheepishly admit that I ask this question knowing next to nothing about algebraic geometry. My motivation comes from this MO discussion where it was clarified that the category of schemes is concretizable (i.e. admits a faithful functor to $\mathbf{Set}$) if and only if it is regularly wellpowered.

Here's what I know: This MO question quotes SGA giving a characterization of monomorphisms *locally of finite type*, but not arbitrary monomorphisms. This MO question indicates that every regular mono is a locally closed immersion. So I would be very happy if someone could tell me whether a scheme can have a large number of locally closed immersions into it.

Locally Presentable and Accessible Categories. – Tim Campion Mar 19 at 14:30