Thanks for the heads up, but I don't think the meter is the problem here. The (absence of a proper) measurement procedure most surely is. I don't think the meter is giving erroneous values on its own. I measured 0.7 mV both on the UT61E and the UT139C and they both agreed.
Right now I am mostly curious about how on earth I was able to get those two series of fast, clean and repeatable measures the first and second night. It is clear that the meter needs some 10 minutes to stabilize the reading, but I am absolutely certain the measures I did in 0°C and 4 (or was it 7)°C were very fast, but they shouldn't have been.
I tried again, same place, same time of the night (but different day and temperature, but most importantly different probes) and I get the same slow rising to the equilibrium value I get inside the house. I wonder if the relative humidity is playing a dominant role in this. Since the measurements are so dependent on small amounts of charges on the base, an environment that does not allow for charge to accumulate or deposit (if not driven) might explain what I saw, but ...
To add to the list of coincidences, here is a plot showing how - in numberland - the simulated BJT VBE with an open base changes with VCE, while stepping the shunt resistor from 1gigaohm to 71gigaohm in 5gigaohm steps. The first curve on the left is the simulation for actually open base at 18°C.
https://i.postimg.cc/jS5kcmz6/screenshot-10.pngThe dots are the measurements I took with 2.9V(*), 4.36V, 8.3V, 16.3V at 19.5°C. Of course there are errors associated with each measure and the actual VBE, like beta, would be different in simulation and actual transistor but... I am getting a trend with VCE that mimicks what should be expected.
(It's true that all exponentials look alike and this might be a different exponential process, but... another coincidence?)
(*) the 2.9 VCE measurement was not exactly reproducible. I was able to get this trend starting from 0 and then going up in voltage, but when I got back (by switching batteries by hand, therefore messing with charges in the worst possible ways) reducing the voltage, the higher voltages were close to the ones shown, but the 2.9V VBE was much higher than that shown.
I plan to redo this measurement with appropriate shielding, clean probes, shorting all terminals before each measurement and in an electrically quiet environment to see if I can get a reproducible curve that fits the expectations. I would appreciate if someone in this forum, with better instruments and setups could try to see if this is a feasible measurement or not.