Depends - do you have 3 phase power?

The size of the cable you need depends on the current that it will carry, voltage drop at that current - which is calculated from the point of supply to your property right to the end of your new cable. So you have to allow for your mains cables and any submains as well. And consider other loads in your house.

And it also depends where the cable is installed - if it is surrounded by insulation, the max current is less, as the cable will heat up too much otherwise. There will probably be other factors as well.

Also for EV charging the above factors are really important, as the cable will be running at the max current that your EV can draw for hours on end. Meaning that if something is not right, you are far less likely to get away with it.

And consider as well that voltage drop in cables is energy lost as heat. Which is also energy that you have paid for. If you have say 10% cable losses compared to 1% then that extra 9% gets added onto your powerbill for no extra benefit. And considering that you will almost certainly be doing alot of EV charging, Those losses can really add up over say 5 years. So trying to save a couple of $$ on cable is often a false economy. Especially as at the time of installation, it is simply a case of choosing cable A instead of cable B.

Of course it probably won't be so economic if a larger cable means having to replace an otherwise working cable.

I own a UK leaf with a 6.6 kW internal charger which I bought a year ago.

For the first couple of weeks I charged using a thick extension cord and limited the EVSE to 10 A.

I then decided to get a new 32A cable put in to a 16A Caravan socket going back to the main switchboard which I charge to 16A max which easily copes with electricity demands of charging my car overnight. The sparky said that if I wanted to change up to 32A draw it would only be about 30 minutes work as he wouldn't need to go under the house or crawl around in the attic and the higher cable only cost a couple of dollars a meter more to install compared with a 16A.

If I do decide to upgrade to use a 32A supply I would need to get a outlet box/ "charging station" which could handle the greater ampage so I would go for something than had a Menzies type 2 socket (rather than one with an integrated leaf cable) as this would potentially allow different types of EV to charge and would have longer life expectancy. My uk leaf came with its original type2 to leaf cable. 

(https://www.nzta.govt.nz/planning-and-investment/planning/planning-for-electric-vehicles/national-guidance-for-public-electric-vehicle-charging-infrastructure/charging-point-connectors-and-socket-outlets/).

How much power do you need to supply to charge? Well I imagine that batteries will get bigger with time but unless you are driving longer distances, trickle charging at night would be good enough as you wouldnt need completely full battery each morning.

ie Leaf might be using 7km per kW and if you only drive 70km per day you would only need to fillup 10kW to replace it so if you plan on plugging in for 8 hrs per night you wouldnt need a huge power supply.

Even the mighty Teslas only home charge at 7 kW..

If you get caught short one day, when I'd head to a DC rapid charger for 20 minute of charge.

No point going more since the whole house will be lucky to have an 80A supply, and more likly a 63A, and when I asked my tame sparky about it, even running a single 32A circuit would probably put me over the theoretical calcs they use for demand sizing. I think I am basically at that limit now with having 2 out buildings.