Rechargeable AA NiMH batteries will work fine (or C or D cells). If you don't mind a slow charge rate, trickle charging them is easy and works. Hook up 6 or 8 cells in series and use a DC-DC buck converter. This is essentially the basis for a UPS I make.
For most all battery types you will have to isolate the battery from the load during a *fast* charge, this is generally the issue with off the shelf lithium charge boards and power banks. You have to switch the load over to your input power (remember the USB powerbank power dip when plugging in?) for the charger circuit to work properly. This is additional circuitry you will have to add that most charger circuit boards usually don't have.
There is lithium iron phosphate too.. generally safer then lithium ion or polymer batteries. Note that Lithium Iron Phosphate and NiMH are pretty close in power density. I'm pretty sure NiMH still wins in price per capacity on consumer type cells.
A constant voltage charge on lead acid batteries will avoid the switching circuity, albeit a bit on the heavy side. For that type of battery you can charge it while in use, since you are just providing a constant voltage and the battery 'floats' to it's full charge voltage and essentially stops charging.
As they say, lots of ways to...
Power problems? MoPower UPS for the Pi