[Ecky]

Be aware that you don't need just amps, but also volts. Tesla uses a 375v nominal system, so you would need 250 AA cells in a series, plus a BMS connected to all of them, to get a 3.0Ah 375v battery pack. I'm seeing about $1.50 per cell, so you're looking at about $375 + BMS, wiring and casing per 3Ah "string" of batteries.

At ~18g per cell, you're looking at roughly 11lbs per string, + BMS, wiring and casing.

I can't find any published spec sheets on discharge rate and I'm no battery expert, but I think 10-15A would be a generous estimate as an upper limit to how fast you can drain these. 10A would give you about 3.7kW of power, or roughly 5HP. If you drew any more, you'd probably kill your battery. Charge current would be a small fraction of this - maybe 2-3 amps inefficiently, and 1-2 amps without turning most of it into heat.

3.0Ah * 375v gives us around 1.1kWh of capacity. You can chop 20% (or more) of that off to preserve battery longevity, so let's give an upper limit of around 0.9kWh per string. A Nissan Leaf is EPA rated to go around 100 miles on 30kWh, or ~3.3 miles per kWh. Therefore, each battery string would be good for around 3 miles in perfect conditions.

To get the same 100 mile range as a Nissan Leaf, you would therefore need 34 strings of batteries, totaling $12,500 in just cells. You could safely regen at around 25kW, or 34HP worth of braking - which seems reasonably useful, but not incredibly fast. It would weigh 375lbs in cells, plus probably another 75-100lbs in wiring/BMS/casing - which is a bit lighter than a Leaf's battery, at least, but it would probably have a fraction of the lifespan.

On the other hand, you could buy 3-4 used Nissan Leaf batteries for the price of a single set of cells for your proposed AA-based battery, and probably 5-6 after you consider wiring it up.