What is the maximum possible current that can mains supply can deliver?
The answer is not infinity, even 'theoretically'.

There are at least 3 defined currents for any point on the supply, and that point can be anywhere from the terminals of a nuclear power station, down through transformers and transmission lines, right to the socket on your wall.

1) Rated current

This is the maximum current it's designed to deliver, 24/7, without overheating. All upstream components are designed to carry this indefinitely.

2) Minimum fault current

This is the minimum current that should flow if you short circuit the output. It is designed to be big enough to open fuses 'quickly', before thermal damage occurs to upstream wiring. It's often used to define the minimum resistance of safety earth bonding, so that in the event of a line to chassis fault, enough current will flow to open the live breaker. It's no good have a (let's say) 10 ohm ground connection, that just sits there and cooks in the event of a short to ground, while failing to open the breaker.

3) Maximum fault current

This is the maximum current that should flow if you short circuit the output. It is designed to be small enough to not damage up stream wiring while the breakers open, and to specify how much current the breakers should be able to stop successfully when choosing the breakers for that link. Damage could occur thermally, or on big feeders through Lorentz forces tearing them out of mountings. This is achieved by having a high enough impedance in the upstream wiring and through transformers, both their resistance and their leakage inductance. Distribution transformers are installed to a minimum impedance specification. It often has to be increased locally to protect areas of the grid. 

Obviously there is an economic tradeoff here. A high impedance supply will generate lower fault currents (easier to protect), but will lose more power in line resistance, and have more voltage drop with varying load. The cost of the supply rises as its quality rises, which is kinda what you'd expect.

It is necessary to have a defined maximum fault current, because short circuits will happen, and replacing installed cable and transformers is far more expensive than replacing breakers. Lightning strikes can initiate arcs, and I once had a very entertaining grandstand view of a digger putting its backhoe through an 11kV feeder on an adjacent building site. Our lights dimmed for about one second before going out, a puff of smoke rose from the hole, as he tripped the breaker to the entire business park.