Thus if charge built up on the input capacitor I would expect it to discharge across the resistance in a second. Even if I'm off by an order of magnitude or two it would still discharge noticeably over the course of a minute.
So it seems to me like some noise in the room or something must be someone continuously building up the charge on the cap.. This is why I brought up the idea of stray capacitance to power sources in the room. Could be an AC signal rectifying on something in the meter.. I am worried I am way off base in my goose chase for what might be creating this signal in the disconnected meter.
It's leakage from the ADC or input amplifier.
This input signal goes to some electronics, those electronics contain various voltages inside, no insulation is perfect, some tiny current "leaks" and charges the input capacitance. Sometimes it drifts towards some specific value, sometimes it just keeps rising until it gets out of range.
There are no gigaohm resistors in those devices either. It's only a declaration that input resistance is no less than 10G or whatever.