Can palettes dock to notebooks?

You can simply assign the content of the palette to the docked cells, however this won't give any nice results for palettes that where not designed to be used as such:

For instance, first open up a palette, I chose Basic Math input, then run

plt = Notebooks[][[1]]
CreateWindow[DockedCells -> NotebookGet[plt][[1]]]

What you then end up with is a notebook with that palette docked, in this case and with most other palettes the result is quite horrible to look at.

Building off of jVincent's idea here's a way to make it take up less space:

plt = NotebookGet[Notebooks[][[1]]][[1]];
CreateWindow[
 DockedCells ->
  ReplacePart[
   plt,
   {1, 1, 1, 2, 2, 1} ->
    Partition[Flatten@plt[[1, 1, 1, 2, 2, 1]], 2]
   ]
 ]

I just looked inside the thing and flattened the primary GridBox.