How many amps can a car battery supply?

First, it highly depends on the battery. Some cars have much beefier batteries, measured in Amp Hours. We arn't even talking about Electric Vehicle battery banks which are massive. Then it depends on the type of battery. Some chemistries are different. Some are 24V instead of 12V. Some cars have more than one. Etc.

That said, the normal peak current is the Cold Cranking Amps. This is the amount of current the battery should provide for starting a cold engine at 0°F. 300 to 1000 Amps is not unusual.

This white paper describes a dead short test:

Finally, each battery was “dead shorted”, connected to a “shorting circuit” consisting of a shunt (5000A+ 0.25%), Hall effect transducer [model LEM LT 4000T (4000A+ 0.5%)], 26 feet of MCM-550 cable and a knife switch. A 2 channel Fluke 190 Scopemeter with automatic triggering was attached to the Hall effect transducer and to the battery terminals. Current and voltage readings were recorded at 0.2 millisecond time intervals from 0 to 0.2 seconds. An Agilent 34970 data-logger was used to monitor the shunt current and battery terminal voltage at 40 millisecond time intervals from 0 to 30 seconds. The “shorting circuit” had a resistance of 1.80 milli-ohms, as measured with a Biddle DLRO micro-ohmmeter. The inductance of the circuit was not measured.

To determine the effect of temperature, sets of UPS12-140 (12V-33AH) batteries were float charged at 13.65V (2.275 volts/cell) for 48 hours at 2, 11, 24, 33 and 40oC in a temperature-controlled environment. OCV, impedance and conductance readings were measured and each battery was “dead short” tested using the test method described above.

In theory, with a perfect conductor you are looking at over 2000 Amps. With their test, they saw 1700 Amps.

And these are just 33 Amp Hour batteries, small compared to most cars. These are UPS batteries! My car has a 150 AH battery with 750 CCA, and it's not even a premium battery.

In short, we are talking about literal Thousands of Amps for a dead short.

But any multimeter worth a damn will have high resistance for voltage measurements, and a fuse for current measurements, that should blow before you see much current flowing. But it will blow if you short it. No if and or buts about it.

Hundreds of amperes. For example, my truck has a battery rated at 625 amps. Each battery should have a rating. Many auto parts stores have the ability to test the battery for you to make sure it is putting out the correct current.

Assuming the battery and the ammeter both have internal resistances of 1 milliohm, then from Ohm's law,

$$ I = \frac{E}{R} = \frac{12V}{0.002\Omega} = \text {6000 amperes} $$