Here is a common situation I run in to:
I have two meshes in the same object that end up near each other and I would like to join them so that the two disparate geometries are linked and manifold. I don't have a common set of vertices between the two objects that I can use
remove doubles on. I think I would usually try to extrude the cube into the larger shape and then use
intersect (knife) with self intersection to get a common set of vertices that I can try to join. However, this is inexact and can yield messy results after the knife, along with leaving extraneous pieces that need to be removed.
So my question is: is there an EASY way to join these together? In the exact situation above, the face of the larger object that faces the cube would be split into 3 faces - one on each side of the cube and the one face shared between the two geometries.
EDIT: I made the mistake of not showing an example that is similar enough to what I actually run into. Consider the following:
There are two answers concerning loop cutting (and extruding) but that doesn't work in the above situation because the work eventually joins back to itself and it's not always a nice square situation when it does.