The car battery must always be fully recharged?

Not necessarily, because lithium batteries on the electric cars can be recharged, even partially, many times and allow the interruption of the recharging process every time you choose, with no memory effect.
Technically speaking, the partial recharging of the batteries is called “opportunity charging”.
Opportunity charging allows the use of many charging columns through the itinerary, for instance at your working place or at the supermarket, increasing the autonomy of your own vehicle, according to your movements.