OK, folks, I'm considering going further than 4x4x4 and entering
the realm of the 5x5x5.

Cubies:
        C := Corner
        X := aXis (center)
        E := Edge (outside center)
        L := Left (external edge)
        R := Right (external edge)
        D := Diagonal (internal [to the face] corner)
        A := Adjacent (to the center, thanks to WER)
                (internal [to the face] edge)

A 3-D view would look like this

                z=-5   +---+---+---+---+---+
                      /   /   /   /   /   /|
                     / C / L / E / R / C / |
                    +---+---+---+---+---+  |
                   /   /   /   /   /   /|C +
                  / R / D / A / D / L / | /|
                 +---+---+---+---+---+  |/ |
        z=0     /   /   /   /   /   /|R +  |
               / E / A / X / A / E / | /|L +
              +---+---+---+---+---+  |/ | /|
             /   /   /   /   /   /|E +  |/ |
            / L / D / A / D / R / | /|D +  |
           +---+---+---+---+---+  |/ | /|E +
          /   /   /   /   /   /|L +  |/ | /|
         / C / R / E / D / C / | /|A +  |/ |
y,z=5   +---+---+---+---+---+  |/ | /|A +  |
        |   |   |   |   |   |C +  |/ | /|R +
        | C | L | E | R | C | /|D +  |/ | /|
        |   |   |   |   |   |/ | /|X +  |/ |
y=3     +---+---+---+---+---+  |/ | /|D +  |
        |   |   |   |   |   |R +  |/ | /|C +
        | R | D | A | D | L | /|A +  |/ | /
        |   |   |   |   |   |/ | /|A +  |/
y=1     +---+---+---+---+---+  |/ | /|L +
        |   |   |   |   |   |E +  |/ | /
y=0     | E | A | X | A | E | /|D +  |/
        |   |   |   |   |   |/ | /|E +
y=-1    +---+---+---+---+---+  |/ | /
        |   |   |   |   |   |L +  |/
        | L | D | A | D | R | /|R +
        |   |   |   |   |   |/ | /
y=-3    +---+---+---+---+---+  |/
        |   |   |   |   |   |C +
        | C | R | E | L | C | /
        |   |   |   |   |   |/
y=-5    +---+---+---+---+---+
     x=-5  -3  -1   1   3   5

LOVE THAT ASPECT RATIO !!!!

All in all there are
	 6 aXis faces
	 8 Corners
	12 Edges
	24 Left/Right type edges
	24 Diagonals
	24 Adjacents
	--
	98 = 5^3 - 3^3 = 125-27 visible cubies

Computation (inaccurate, but within a couple orders of magnitude)
of the number of reachable positions:

Axes:	lets not hack the extended problem yet	->	1
Corners:8 of them anywhere			->	8!
	each can take 3 orientations		->	3^8
	parity of the corner			->	1/3
Edges:	12 of them anywhere			->	12!
	each can take 2 orientations		->	2^12
	position/orientation restriction	->	1/4
L/R:	24 of them anywhere			->	24!
	orientation defined (they cannot flip)	->	1
	parity (cannot swap only two)		->	1/2 (I think)
Adjac:	24 of them anywhere:			->	24!
	one edge always touches a face center	->	1
	parity					->	1/2 (at least)
Diags:	24 of them anywhere			->	24!
	one corner always touches a face center	->	1
	parity					->	1/2 (at least)

It may not be accurate, but this computation gives
	1.291318 * 10^90

A slice through the center (z=0) probably looks something like

y=5\
   /    ..XXXXXXXXXX++++++++++EEEEEEEEEE
        ..XXXXXXXXXX++++++++++EEEEEEEEEE
        .....XXXX++++++++EEEEEEEEEEEEEEE
        .....XXXX++++++++EEEEEEEEEEEEEEE        X is an axis cubie
y=4\    .....XXXX++++++++EEEEEEEEEEEEEEE        E is an edge cubie
   /    .....XXXX++++++++EEEEEEEEEEEEEEE        + is one adjacent cubie
        .....XXXX++++++++EEEEEEEEEEEEEEE        ~ is another adjacent
        .....XXXX++++++++EEEEEEEEEEEEEEE
        .....XXXX++++++++EEEEEEEEEEEEEEE
y=3\    .....XXXX++++++++EEEEEEEEEEEEEEE
   /    .....XXXX++++++++EEEEEEEEEEEEE~~
        .....XXXX++++++++EEEEEEEEEEEEE~~
        .....XXXX++++++++EEEEEEEEEEEEE~~
        .....XXXX++++++++EEEEEEEEEEEEE~~
y=2\    .....XXXX++++++++EEEEEEEEEEEEE~~
   /    .....XXXX+++++++/~~~~~~~~~~~~~~~
        .....XXXX++++++/~~~~~~~~~~~~~~~~
        .....XXXX+++++/~~~~~~~~~~~~~~~~~
        \....XXXX++++/~~~~~~~~~~~~~~~~~~
y=1\    .\...XXXX+++/~~~~~~~~~~~~~~~~~~~
   /    ..\..XXXX++/~~~~~~~~~~~~~~~~~~XX
        ...\.XXXX+/~~~~~~~~~~~~~~~~~~~XX
        ....\XXXX/~~~~~~~~~~~~~~~~~~~~XX
        XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX
y=0\    XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX
   /    XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX
        XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX
        ..../XXXX\....................XX
        .../.XXXX.\...................XX
y=-1\   ../..XXXX..\..................XX
    /
         /\   /\   /\        /\        /\
       x=-1    0    1         3         5

This time the central axis is rigid in the sense that it does
form a cross, but each of the six spokes can rotate as in the
3x3x3 cube. The curvature and tolerances of some of the pieces
gets a little hairy, but I'm working with graph paper and looking
at the other slices through the cube. Wish me luck -- I have
thoughts of construction.