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This is a test of concept & implementation for a layered, ball-bearing  maze-type puzzle.  I think a wood version would be great, but I had to test the idea first. 

Each layer of the egg puzzle has some holes, which can go straight down, curve and go down or only go part way through to the other side.  Each layer also spins around a center pole.  The goal is to get a ball through all the layers of the egg.  In the default position (slices aligned to form nice egg shape),the ball can only go through one layer at a time. The player inserts a ball into the top layer, twists that layer to the right or the left, until the ball falls through to another layer.  (Metal balls work best because you can feel and hear them fall through the layer.)  Twist the each layer  to get the ball down to the next layer, until the ball rolls out of the bottom layer.  Some holes are 'traps' and do not let the ball pass through, if this happens, invert the puzzle and go back up a layer to try again.

I used
blue styrofoam insulation (2 inch foam insulation board)
a sharpened wood pencil
a thin-bladded, serrated knife (pumpkin carving knives work well)
a hack saw (optional)
drill or cork hole cutter
acrylic craft paint
small metal ball bearing (a small 'steelie' marble)

Step 1: Overview

The egg shape is sliced into layers.  A hole with a specific radius from the center of the layer can line up with a hole in the next layer, as long as both holes are at the same position.  In the diagram, hole A in the top layer is at the same radius as hole A in the second layer.  From the start position, the top layer is rotated clockwise until both holes align and the ball falls into the second layer.  The hole in layer 2 is cut at a slant and ends at a different radius from the top of the hole.  If layer 2 is twisted clockwise, the ball will fall into hole A in layer 3, which is at the same radius as the bottom of hole A in layer 2.  Turn layer 3 clockwise and the ball will fall into hole A on layer 4.  Turn layer 4 counter-clockwise to reach hole A in the bottom layer.  The holes in the bottom layer slant so the ball will roll out of the side of the egg, giving you a chance to catch the ball before it hits the floor.
 
The hole at position B has a variable path.  Turn the top-counter clockwise until both B holes align and the ball falls to layer 2.  Turn layer 2 counter-wise and the B holes will align, dropping the ball to layer 3.  Layer 4 has a trap.  If you turn layer 3 counter-clockwise, hole B will align with a grey hole at radius B.  This hole only goes halfway through the layer and the ball is trapped.  If layer 3 is turned counter-clockwise, the ball will fall into the while B hole on layer 4.  Rotate layer 4 clockwise and the ball will roll through hole B in the bottom layer, and out the side of the egg. 



Step 2: Make the Egg Shape

Find some good foam - 2 inch rigid styrofoam insulation works well.  You could also use florist foam or other fine grained styrofoam.  Draw an oval shape and cut it out of the foam.  Shape the foam into a 3-d egg shape.  A wood file works very well, just remember that files only 'cut' in one direction, just push down on the away stroke.

Step 3: Slice Egg Puzzle

Prepare your egg for slicing.  Carefully push a sharpened wood pencil through the center of the egg, this will be the center axis.  You can start the hole from both ends and meet in the middle.   Watch your fingers - a pencil point stab hurts :-)

Remove the pencil and evenly slice your egg into 4 or 5 layers.  Each level should be at least twice the diameter of your ball. three or four is better.   Do not copy my poor slicing, keep your slices even, and your puzzle will work more smoothly than mine.  Use a hack saw or your serrated knife.  If your edges are rough, lightly sand the flat sides of each slice.
 

Step 4: Cut the Holes for the Steel Ball to Roll Through

Plan out your holes, you can use the overview figure as a model.  Some things to remember:
In the default egg shape, a hole should drop only one level (slice).  Holes should line up with a lower level only when you rotate a level. 
Keep at least a hole's width between holes on each level, so you have a structurally sound egg.
Decide how many 'traps' you want to set.

Once you have a plan, cut or drill your holes.  I found a cork cutter in the garage and it worked great on blue styrofoam.  It's basically a metal tube that is sharpened on one end.  Press and twist it to cut a hole.  You can cut holes straight down or on a diagonal.  You can also cut 'elbow' holes, cut at an angle from both the top and the bottom until the holes meet form a 'bent hole'.  Use a small, round file to clean up the holes.

Traps are holes that only go partway through the level, lit little pit traps.


 

Step 5: Paint Your Egg

Insert the pencil in the middle.  Cut pencil to height of the egg.  Decorate you egg puzzle with any painted design.

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