The finity machine is a type of computer with infinite precision and finite power. It may serve a variety of purposes. From observing a universe to being a simple home appliance. 
In the story a Finity Machine is the "diegetical fourth wall"; that is, the device within the universe of Stranger Eons by which the readers are able to read and interact with the story. Countless Leaves has demonstrated to have some control over it in C1I1 "Ego Maintenance".