Infinitely Rich Objects In A Topos In the category of graphs I think one can find a graph $X$ that has a subgraph isomorphic to any finite graph $A.$ 
My question is, can this type of situation be generalized to a more general topos ? And if so, how ? In particular, is there an appropriate notion of an object being "finite" such that the following definition (1) makes sense, (2) refers to something existent, and (3) generalizes the graph case above:
We define an object $X$ in a topos to be infinitely rich when, for any "finite" object $A$, there is a monomorphism $ A \rightarrowtail X.$
Perhaps this kind of idea has already been studied, and if so I would like to know. Anyway, it looks like there are several different notions of finite objects in category theory. What I want is to select an appropriate idea of a "finite" object, so that an infinitely rich object can be found in my topos. My basic question is how to do this. 
One idea I had is to suppose the topos is cocomplete, and then let $X$ be the coproduct of all the finite objects. However this requires that the finite objects form a small set.
 A: In any locally presentable category, the isomorphism classes of compact objects (that is, objects Homs out of which commute with filtered colimits) form a small set. This is a fundamental fact about locally presentable categories which can be found for instance in  Chapter 2 of Adamek and Rosicky’s book on the topic. Then your solution of taking the coproduct works great. Incidentally, this is probably the finiteness notion you want, as it coincides with the natural notion in any example of a topos you might name.
Luckily, “most” cocomplete toposes are locally presentable. In fact, Grothendieck toposes are precisely the locally presentable toposes. I cannot immediately produce an example of a cocomplete topos with a proper class of non-isomorphic compact objects. If $G$ is the large group given as a coproduct of representatives of every isomorphism class of finite group, then there is a cocomplete non-Grothendieck topos of small sets with a $G$-action, but it has only a small set of isomorphism classes of compact objects.
