Does a graph with infinite nodes that is not fully connected have a countably infinite or a uncountably infinite number of paths originating from a single node?
We are only concerned with paths that do not contain loops. Also, we can assume that each node has between one path to the maximum possible number of unique, non looping paths to each other node.
If I take a graph with vertex set $\mathbb{R}$, and connect 0 to every $x \in \mathbb{R} \setminus \{ 0 \}$, then I have uncountably many finite paths from 0.
If on the other hand you require the vertex set to be countable, then the number of finite paths from any given point is countable. (As it is a countable union of countable sets).
As Tryss says, if you allow paths to be infinite in length, then the cardinality of paths extending from a point is heavily dependent on geometry.