Stock '66 Chrysler 300 w/ 383 4bbl. When I turn the key to start the engine, it makes a click and the whole car loses power. Disconnect battery and reconnect, car has power. Try to crank and same thing happens. Have tried 2 different batteries, one of them brand new. No luck

Any ideas?

Battery cables loose or dirty on one end or the other. Check the cables at the starter relay and starter also.

How is the Battery itself?

Batteries are fully charged - work fine in my '66 Fury III

All connections are clean and snug. Could it be a bad starter relay?

Don't see how a bad starter relay would kill power to the entire car.

... and the ground(s).

Will check all grounds in the morning. It is very strange how disconnecting then reconnecting the battery restores power to the car.

That is where the problem is, in the connections at the battery terminals, or where you are flexing the cables moving them to changing the batteries.

What Nanket said & if good in that large cable system then next time it goes dead see if you have fire at the alt "batt" stud & you might seperate/check/clean the bulkhead and check the ammeter terminals in that path. For it to be completely dead there would need to be an open between the batt positive post to/thru the bulkhead & to/thru the ammeter to the "welded splice". Bulkhead/ammeter are culprits & with it going completely open/dead you'll easily see it (poor/loose connection/terminal) with some checking

If the battery cables are old, sometimes they turn into that nasty green stuff or corrosion under the insulation. They look good from the outside, but look like crap inside. Someone posted something similar, but changed out all his battery cables and all was good.

I agree. I have had a bad ground cable do something similar.
it wouldn't start at all.
crushing the cable by bending it 90 degrees at the block got it running again, but I swapped it out and all was good after that.

you can prove it to yourself by running a new seperate ground to the block and checking your lights come back without disconnecting the battery.
use a good large gauge wire if you have it.

Thanks for the help guys - long story short it was a bad ground. The Block/Body cable had corroded in half! Replaced that and cleaned up the battery/block connection and all is well again.