By the way, are you sure about that 3.7V? That's the nominal value of a LiPo battery but the real output is usually somewhere between 3.2V and 4.2V depending on the charge. Did you measure that with your multimeter?
I did a measure with the multimeter on the + of the battery and tells me the battery is providing 3.7 volts, this could be fine because the battery isn't fully charged.
Not ALL microcontroller ADCs have high input impedence, and they CAN be affected by using a series resistor with too high a value. IIRC, some of the 8bit PICs have an input impedence of ~10k - using a 100k series resistor would be a very bad idea on one of those.
So the situation where im in now is not ok? Using a 100k resistor between the + and the A0.
Here is an example which includes a battery voltage monitor. Interestingly, it includes a pinMode() statement for A0.
pinMode(A0, INPUT);
Usually, this would not be necessary but maybe worth a try.
Certainly, based on your attempts to use the VCC mode, the ADC is still working.
I also tried the INPUT way but doesn't make sense. As PaulRB said, the most sketches i saw doesn't have a Pinmode either.
BTW nice to hear the ADC is not broken, so there is something else why the output is always the same.
Another thing i was wondering:
Does a connected USB to the laptop affect the reads?