How big an antenna would a cellphone with an AM receiver need?

AM is around 1Mhz so that's a 300 meter wavelength. For a half wave dipole (basic high efficiency antenna) that's 150 meters of wire. A vertical antenna (like on your car) using a ground plane as the other end of the dipole would be 1/4 wave so 75 meters.

All this said, AM @ 1Mhz propagates very efficiently regionally and they broadcast at insane power levels so efficiency on your end isn't that big a concern if you're within a couple hundred miles of the transmitter. You can get away with a meter long antenna just fine.

If I remember correctly, cellphones with radio receivers tell you to plug in headphones whenever you're tuning in. My guess is the headphones cables act as the antenna.

[Edit: TIL AM stations seem to transmit at frequencies way smaller than FM]