Easiest Way to Join Meshes 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.
Since you are asking about the "easiest way"...

In my opinion, the easiest way to deal with those situations is by not creating them in the first place (unfortunately minecraft and other apps seem to have created unnecessary problems when it comes to modeling, by thinking in boxes that get added and somehow have to be connected. You have to think outside the box...).

If you know that those parts have to be connected, create them as connected elements from the start. 

In this case create a base mesh, scale, add a couple of loop cuts and extrude. 

The shape of your object is the result of it's topology. Topology shouldn't be an afterthought