Here's another example, inspired by aws's but not involving any computability. Let $A$ be a distributive lattice and let $\mathrm{Fam}(A)$ be the free coproduct-completion of $A$, whose objects are families $\{a_i\}_{i\in I}$ of elements of $A$ and whose morphisms $\{a_i\}_{i\in I} \to \{b_j\}_{j\in J}$ are functions $f:I\to J$ such that $a_i \le b_{f(i)}$.

Then $\mathrm{Fam}(A)$ has finite limits (given by limits in $\mathrm{Set}$ and meets in $A$) and pullback-stable finite unions (given by unions in $\mathrm{Set}$ and joins in $A$; distributivity of $A$ is needed to make these pullback-stable). Its monomorphisms are arbitrary morphisms whose underlying set-function is injective. But it doesn't have images unless $A$ is a complete lattice; the image of the unique map from any family $\{a_i\}_{i\in I}$ to the terminal object would have to be the one-element family on a join of all the $a_i$.

Here's another example, inspired by aws's but not involving any computability. Let $A$ be a distributive lattice and let $\mathrm{Fam}(A)$ be the free coproduct-completion of $A$, whose objects are families $\{a_i\}_{i\in I}$ of elements of $A$ and whose morphisms $\{a_i\}_{i\in I} \to \{b_j\}_{j\in J}$ are functions $f:I\to J$ such that $a_i \le b_{f(i)}$.

Then $\mathrm{Fam}(A)$ has finite limits (given by limits in $\mathrm{Set}$ and meets in $A$) and pullback-stable finite unions (given by unions in $\mathrm{Set}$ and joins in $A$; distributivity of $A$ is needed to make these pullback-stable). Its monomorphisms are arbitrary morphisms whose underlying set-function is injective. But it doesn't have images unless $A$ is a complete lattice; the image of the unique map from any family $\{a_i\}_{i\in I}$ to the terminal object would have to be the one-element family on a join of all the $a_i$.