I don't know why this keeps coming up, but it does.
The Tesla Model S 85kwh battery pack has a 74P96S battery pack configuration (sets of 74 18650 cells in parallel, then 96 of these sets in series).
Fully charged, the Model S pack is right around 400v. At 4.2v/cell, this should be 403v, so I'm happy to call "fully charged" 4.2v.
Nominal voltage on the cells will be around 3.7v/cell, or 355v.
The cells are each ~3AH (3000mah).
At highway cruise speeds, a Model S manages roughly 300WH/mi at 70mph - or 21kW. For a fully charged pack, this is 52A, and at nominal voltage, 59A. This correspond to 0.7A or 0.8A per cell, for a C rate of ~0.25C.
Peak power consumption on the P85D is about 470kW. This corresponds to full/nominal amperages of 1175A/1324A, for a per cell current of 15.8A/17.9A, for a C-rate of 5.25C/6C (briefly - the car will hit max speed in a hurry).
The normal P85 cannot sink so much power, so peak C-rates are lower.
Ludicrous mode seems to peak around 570kW (based on Wikipedia). This requires the 90kwh pack (or roughly 3.2AH individual cells, assuming the same pack configuration). This corresponds to full/nominal amperages of 1415A/1605A (though I understand it's limited to 1500A).
1500A out of the P90 pack works out to a per-cell current of 20.3A, for a C-rate (on the 3.2AH cells) of 6.35C - so slightly more than the existing pack, but not by much.
Talking to some Tesla owners, the cars only accelerate "at their best" when fully charged - so the amperage being limited and acceleration being slightly worse when drained sounds like that's actually how things work. This being, of course, utterly opposite from an internal combustion vehicle, which will accelerate best with a nearly empty tank (less weight).
Still very, very impressive!