Rounding off a single corner of a cube There's many guides and approaches to rounding off edges or multiple vertexes of a cube. But I have not found a resource showing how to take a single corner of a cube and beveling it.
The closest I have come to reaching my goal is shown here. 

I want the top side of the cube to have one rounded corner while the opposite face remains square.
The second image is taking from the cubes bottom side. It have become rounded at one corner, while I wanted it to remain square.

The remaining sides of the cube must be rounded off accordingly too. Providing a nice rounded corner, that can be placed upon another cube and have all their edges touch.

I'm still learning Blender so forgive me if I have missed the obvious.

Edit: To try and illustrate what I am aiming consider the two red blocks. The bottom side of the blocks are squares. They have however been rounded on each left and the back side. The nearest corner of the cubes seem to have been smoothed out.
I think that the "smooth" look of the corner is just a matter of shading (and shaders).

I suggest to simply model it as a cloister vault's corner. Any distortion of the profiles in my opinion would lead to a worse result.



If your boundary profiles are round, you can use the Bevel operator on the two top edge (shortcut Ctrl+B). Then add two (flattened) loopcuts near the corner and Remove Doubles.



Then control the shading (set it to smooth) by marking sharp edges  and adding an Edge split modifier as you can see in the first picture.