What is the easiest way to stagger a signal chain on a stack of identical boards?
What kind of analog signal?

One technique that works well is to convert the analog signal to current.  You use current sources on each of your boards (a proper current source has very high output impedance) and simply sum all of your currents together onto a common bus.

You receive the sum of all those currents at one location.  The receiver can be a proper current-to-voltage converter or a simple resistor.

For what it's worth, most professional audio intercom systems work this way.  You can have a hundred or more users on a single intercom line and adding or removing users does not affect the audio level significantly.

The current source that is used for audio intercom systems is usually some variation of the Howland Current Pump (also known as a Howland Current Source).  There are many resources available on the web to show you what it is and how to use it.  

I first saw this circuit described many years ago in a National Semiconductor app note where it was called a "bilateral current source".  I promptly adopted it for the intercom system that I was building for myself at the time.  It was quite some time later that I learned of the proper name for the circuit.

The advantage of this system is that the input pin is electrically connected to the output pin.  That means that you can stack boards on top of each other and use the same pin to collect the signals from all the boards.