I recently ran across several threads on the Backyard Chickens forum about weird eggs laid by hens.  It turns out that these are not terribly uncommon, which is a good thing to know ahead of time!

First, some chicken biology on how an egg is formed.  A chicken's ovary is like a cluster of grapes hanging over a funnel.  When one of the "grapes"(follicles) matures into an ovum, it falls off the ovary and drops into the funnel.  From there it travels down a long tube (the oviduct). 

The "grape" (ovum) is the center of the egg, which becomes the yolk as the egg develops.  As it travels down the oviduct, the other layers of the egg are laid around it.  Imagine a car moving through an automatic car wash.  Instead of getting a rinse, a soap and a wax, the yolk gets the egg white, then the membranes, and finally the shell. 

Odd things can happen along the way.  Most of us never see the results of these mis-steps, because we buy our eggs from the store.  It's not like Lucerne is going to stick these weird eggs into your 12-pack!

There are three fundamental ways that something can go wrong:

1.     Air ball.  Not all basketballs make the hoop, and sometimes an egg misses the funnel.  When this happens, it falls into the hen's abdominal cavity.  In some cases it may eventually make its way back into the funnel.  In other cases it can be reabsorbed by the hen's body, or it can get infected and cause problems.

If you find a freak egg that smells very bad, it could well be one of these.  If an egg wanders around the hen's abdomen before falling back into the funnel, it can be pretty rank.  Hens which chronically miss the funnel are called "internal layers," and have to be culled or euthanized.

2.    Wrong direction.  The egg moves along the oviduct through peristalsis, which is a series of muscular contractions.  This is similar to the way that food moves down your throat and through your digestive tract.  And just like eating dinner, sometimes something can go wrong, and the peristalsis can get thrown into reverse.
If we go back to the car wash analogy, during a normal car wash your car goes Rinse -> Soap -> Wax -> Dry.  Imagine if you threw your car into reverse partway through.  It would go Rinse -> Soap -> Wax -> Soap -> Wax -> Dry, and get two coats of soap and wax.  The same thing can happen to an egg.

This is how you can end up with a perfectly formed egg inside a membrane.  Occasionally a chicken will lay a perfectly formed egg inside a second eggshell.  Weird egg, indeed!

3.    Missed a step.  Back at the carwash, imagine if you stepped on the gas and sped through the wax stage.  Or if the waxing machine happened to be broken for some reason.  An egg can slip through without an egg white, without membranes, or perfectly formed but without a shell. 

Sometimes one or more of the steps is permanently broken, or turned off due to illness or stress.  It's not unusual for the occasional egg to miss a step, but if this happens very often you may want to consult a vet, or think about culling or euthanizing the hen.