I went through the gitter conversation. I agree that it is essential to come up with a protypical example where J-holomorphic curves converge to a genuine lamination (Else, as you point out, there is nothing much gained by passing to laminations).
Now I am thinking of examples in CP^2 (in general CP^n) to see if there is such an example. I am looking at curves with varying genera so that one avoids bubbled curves. I will post it if I see something new.