Second floor apartment stairs is what I would guess, especially based upon the different brickwork below the second floor replacement window and incongruous side gambrel dormer. That poor Greek Revival has seen a great deal of remuddling in its time.
Preservation in Pink 
So true!
Looks like it was attached to another house at one time. Or perhaps it had a “wing” on that side.
Second floor apartment stairs is what I would guess, especially based upon the different brickwork below the second floor replacement window and incongruous side gambrel dormer. That poor Greek Revival has seen a great deal of remuddling in its time.