Our design engineer who designed the OpenLogger did an end-to-end analysis to determine the end number of bits of the OpenLogger. This is what they ended up doing in a summarized fashion:
They sampled 3 AAA battery inputs to the SD card at 250 kS/s and set the OpenLogger sample rate to 125 kS/s and then took 4096 samples; they then took the raw data stored on the SD card and converted it to a CSV file and exported the data for processing.
Their Agilent scope read the battery pack at 4.61538 V and as they later found from FFT results the OpenLogger read 4.616605445 V, leading to a 0.001226445 V or ~1.2mV difference, which is presuming the Agilent is perfect (which it is not), but it was nice to see that the values worked out so closely.
They calculated the RMS value of the full 4096 samples in both the time domain and using Parseval's theorem in the frequency domain as well, both of which came up with the same RMS value of 4616.606689 mV, which is very close to the DC battery voltage of 4616 mV.
Because RMS is the same as DC voltage, this gives the previously mentioned DC value of 4.616605445 V. They can then remove the DC component from the total RMS value to find the remaining energy (the total noise, including analog, sampling, and quantization noise) of the OpenLogger from end-to-end.
With the input range of +/- 10V input, this produces an RMS noise of 1.5mV.
At the ADC input, there is a 3V reference and the analog input front end attenuates the input by a factor of 0.1392, so the 1.5mV noise on the OpenLogger is 0.2088mV at the ADC. With the 16 bits (65536 LSBs) over 3V, 0.0002088V translates to ~4.56 LSBs of noise. The ENOB is a power of 2, so log(4.56)/log(2) results in 2.189 bits, giving us a final ENOB of 16 - 2.189 = ~13.8 bits.
Note though that this ENOB of 13.8 bits is based on system noise and not dynamic range, so for non-DC inputs (which will likely be measured at some point) the end number of bits is not easily determined. The datasheet for the ADC used in the OpenLogger (link) shows that the ADC itself gives an ENOB of about 14.5 bits at DC voltage (so the 13.8 bits is within that range), but at high frequencies, this of course rolls off to lower ENOB at higher frequency inputs. Thus, they cannot fully predict what the compound ENOB would be over the dynamic range, but they suspect it all mixes together and is 1 or 1.5 bits lower than the ADC ENOB response.
Let me know if you have questions or would like to see the non-abbreviated version of his analysis.