How would I go about adding some sort of recurring infinite plane?
Something like in this; Discover and Play Free Online Games on Kongregate!
Is it possibly in Unity?
Bump
You could scale your plane to be absolutely huge, and then just set the material tiling to it looks right based on the plane size.
Or you could place a collider border near the edge of each plane, and as your player hits the collider, instantiate a new plane in the direction the player is going.