The primary limits in order from launch are: max torque, max power, and the motor limits at power probably governed by Back EMF.
It’s likely that the limitation of the vehicle right now is the motor max current (torque), not the battery.
This is mostly correct. The limit is the torque that the motors can produce. But the primary thing stopping them from increasing torque further is not battery current, nor is it mechanical limits like gear or axle stress. The limitation here is magnetic flux density inside the motor.
For an induction motor, torque produced in the motor (T) is directly proportional to the magnetic flux density in the stator-rotor air gap (B), the rotor current (Ir), and the rotor phase angle (theta). The magnetic flux density is proportional to the stator current, so long as the core material that the motor is made of is not driven into magnetic saturation.
Basically, for a given cross-sectional area of magnetically permeable material, there is a maximum amount of magnetic flux (therefore a maximum flux density) that can be produced in that material. For the stator windings, we use the inverter to send current through them, that creates a magnetizing force (H, in amp-turns or Webers) which then causes a magnetic flux (B, in Teslas) in that cross-sectional area (A). We can increase H and therefore increase B, up to a certain point where the material cannot carry any more magnetic flux. This point is refered to as "saturation", where an increase in H no longer causes an increase in B. For highly permeable silicon steel, the maximum flux density is about 1.6-1.8 Teslas.
Here are the magnetization curves for 9 different materials. Material 2 (silicon steel) is most common. You can see that as H increases, you get a sharp rise in B until about 1.5 Teslas, then the curve flattens out. You want to operate in the H range of 0-20 amp-turns/in.
If you want more total magnetic flux (therefore more torque) and you're close to saturation, then you can't get it by increasing current. You have to increase the area, which means a bigger motor.
Take a look at the two different sized induction motors in the pre-Raven performance Model S. You can (briefly) use the inverter to send just as much current through the small front motor as you send through the large back motor, but that front motor won't produce the torque that the back one will. That's because you're driving the front motor into magnetic saturation, so you're not increasing the magnetic flux. The rear motor is not near saturation, so the total flux you can create in the air gap is a lot more, that means a lot more torque.
The PMSR motor works differently and uses different equations than the induction motor, but the principal is the same. For the motor's given size, there is a maximum torque it can produce before you can no longer increase the magnetic flux density. At that point, to get more torque you simply have to increase the motor dimensions so that you have more cross sectional area for the magnetic flux to circulate in.
