Take all finite subraphs of your infinite graph, now forget it and ask yourself whether you can recover the graph you started with just from this bunch of finite graphs or not? There is a precise and very simple answer, which is actually delivered by a general method of taking limits in certain categories called Fraisse construction.
For example, the limit of all finite graphs is exactly the random graph, or say limit of all finite linear orders is the dense linear order (like in rationals).
But you can apply the same procedure to groups, partial orders, metric spaces, fields and whatever else getting lots of fancy objects, sometimes well-known and sometimes totally new. I will show a couple of exotic species hopefully.