VCC is the output of the on-board Voltage regulator. You should only (ever) connect a regulated supply Voltage to VCC; 5V for a 5V board (3.3V for a 3.3V board).
To power a Pro Mini directly from a battery, you should connect the battery +V to the RAW pin. The RAW pin is the input to the on-board regulator. You can connect up to 12V to the RAW pin and VCC will remain at a constant voltage, until the input Voltage falls below VCC + the regulator drop out voltage.
For example, if you connected a 9V (PP3) battery to the RAW pin on a 5V Pro Mini. VCC would remain at 5V until the battery discharged to 5.7V. As the Voltage drops further, the Pro becomes increasingly erratic.