Call two nodes $v$ and $w$ of a graph infinitely connected iff there is an infinite path $P(v)$ starting at $v$ and an infinite path $P(w)$ starting at $w$ such that there is an $x \in P(v) \cap P(w)$ which is not connected (in the standard, i.e. finite sense) to one of $v$ or $w$.
It seems to be a simple matter of fact that there are no graphs with infinitely connected nodes, since every $x$ on a simply infinite path is finitely connected to the starting point.
On the other hand, compare the situation with geometry: In standard (i.e. Euclidean) geometry, no two parallel lines do intersect, opposed to projective geometry where all pairs of lines intersect.
Is there something like a "projective graph theory" with nodes "at infinity"?
The concept of infinitely connected nodes might be vacuous in such a graph theory, since all nodes might be connected, finitely or infinitely. But this would probably depend on the theory.