Asking what point to ground it at, is probably already the wrong question; but showing why, unfortunately isn't so easy.
For simplicity's sake, just pour ground over everything. If it's 2-layer boards, add vias to stitch top and bottom grounds, particularly near component pins/pads, and at trace/bus crossings. Maximize the area where signals/components sit above or below a ground. Note that, anywhere traces or rows of component pads cross, a negative space is created through both pours, where ground is not adjacent to them. Minimize these areas, and stitch around them to close the area of that loop.
For 4+ layer boards, just pour GND/VCC inner layers, and route signals on outer layers without much concern. Do avoid placing tight rows of vias, which create a slot in the planes -- give space every 4 vias or so, so planes can pour between them.
This largely solves issues of EMC: signal quality, susceptibility, emissions.
Then, address the DC/signal accuracy issues. If you find too much voltage drop across the plane (likely some ~mV), route the signals between areas differentially, adding differential sense amps as needed. That is, bring along the ground reference, but do it as a net tie or 0-ohm jumper to the reference point, and reference all your signal chain to that. Add filtering (RCs?) as needed to maintain signal quality / stability. This obviates the need for full system star grounding, which is likely to be an EMC disaster.
You're not using the internal ADC, right? That's a, let's see here... Ah, up to 16 bits, not bad. So, if true 16 bits is achievable at all (by the device itself, in the best of circumstances), some care is probably required. And, I doubt that you will have good enough conditions through a stack of dev boards, regardless of how good your customer stacker is. Build a dedicated board, don't worry about doing it this way.
The next best you can do, with the dev board stack as-is, is differential ADC measurements. Either actual differential, or simultaneous sampling (probably an option, if it's so equipped, I didn't check), or pseudo-differential (sample one channel after the other, subtract in SW) -- in order of preference. You may need to add filter caps to the ADC pins, near the MCU, to keep bandwidth (and noise) down at the point of measurement.
If you'll have current flows in various directions, and multiple grounded (earthed) signal sources/sinks, ground loop is inevitable. The best you can do here is isolate each input channel (or set thereof), using probably a separate ADC (and maybe MCU) on the high side to communicate the readings via digital isolator. The second best is to use differential inputs, preferably actually balanced differential, but unbalanced (coax, RCA jack, whatever) can be used in the same way as well, specifically using a low impedance (but not zero) ground-return path, so that EMI is shunted (bypass caps between jack/shield and circuit GND) while some DC/signal offset is allowed through (dropping across a modest value (say 100R) resistor, but limited by TVS diode, say). Use a differential receiver and the signal will come in with good quality, as long as the common mode range is not violated.
Tim
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