What is the most compact design of down-shifting double-pistons?

It is often useful to stack pistons to shift blocks by more than one space. As an extended piston can not be moved, the timing of the corresponding redstone circuit is of crucial importance. Additionally, for the construction of larger gates, you have to line up several of these double extenders.

What is the most compact design of down-shifting double-pistons in a row?

There is none

There is a big stonking problem with the double-extender. It doesn't work vertically. At least not, if you want multiple of them after one another.

In order to line up the pistons, only one side may be used to power blocks adjacent to the pistons. So, let's look at what our redstone has to do when, and where it can do it.

For a down-pushing double-extender to work, we need to take a look at the following schematic:

In order to "close the gate", we need to extend the upper piston first, by powering the top block:

Then, we extend the lower piston, by powering the bottom block:

With this, the double-extender is fully extended. Now we need to pull it back in. For that, we first retract the lower piston by powering down the bottom block:

Now, retract the upper piston by powering down the top block:

Ooops! Now we have a floating block.

To fix that, we need to extend and retract the lower piston again, by powering the middle block:

Good, now we have the theory down.

In practice, this cannot be done on just one side (which is the requirement for having multiple of them in a line). It is impossible to power all three layers individually at these distinct times, because redstone takes up a vertical air block.

A condensed one-sided Double Extender only works horizontally (as taken from Minecraftwiki.net:

Vertical double-extenders only work with powering from multiple sides. For that purpose, I have also built an upside-down version of this vertical double-extender (again) from Minecraftwiki.net: