What's the largest battery I can run?

[Nocte]

Buy 4 SAWs. Strap pairs of them together. Build a frame that will allow you to hold all 4 AEGs strapped to your arms/shoulders. Use a transformer to step the car battery's voltage down to ~9V (this will increase your already substantial current by 25%, too). Wire all 4 SAWS to the battery. Fire everything at once (never one at a time).

Nothing but your bank account and maybe your knees should break.

Congratulations, you are now a tank. Or a giant immobile target, but whatever.