As I was poking around looking at rechargeable battery options, I found
"639-LS14500CBA" batteries over at Mouser with the following specs:
Lithium Batteries 3.6V AA 2700 mAh BUTTON
Since they are not listed as protected, I'm guessing they don't have PCBs. However, I haven't seen any other AA size Li rechargeables rated that high on mAh anywhere else - so I'm guessing it's probably lower.

Has anyone tried these or something similar?
(I did try a search)