Based on the description on their site, the controller includes a built-in battery: "8.39 Wh Li-ion battery, 35+ hours of gameplay... "
That was disappointing for me. Specially condidering the Steam Frame's controllers make use of AA batteries: "One replaceable AA battery per controller, 40hr battery life"
AA Batteries might not be as convenient to use, but being able to replace them is a great advantage. All my Xbox360 controllers still work fine, but none of my PS3' Dualshock 3s.
The official docking station could be used to recharge (rechargables) AA batteries so the functionality could remain the same.
Personally I only always buy those enegizer alkaline batteries. Didn't know you could recharge them, might try next time. I've almost usually always chuck them in the bin.
You cannot charge those, please do not try unless you like the smell of burnt chemicals or a house fire (or both), alkaline batteries are not rechargeable. There are lithium rechargeable batteries with the same form factor (AA, AAA, etc.) or if you use older ones there are also NI-MH or NI-CD (really old).
Just to clarify, the alkaline batteries like below are not rechargeable, and if you try you will have a bad time:
Energizer have their own range of rechargeable batteries, that look like this:
If you have an IKEA nearby, their LADDA (I think?) branded rechargeable batteries tend to be a quite affordable entry point.