The canonusa.com website says that the 580EX speedlight can use AA alkaline, lithium or rechargeable NiMH batteries.
The website gives no information about the CP-E3 battery pack.
If you assume that it can use the same batteries as the 580EX speedlight, your only choice for rechargeable batteries is the NiMH batteries.
These are available (with charger) at any store that sells cameras.
I would buy a brand that belongs to a nationally known battery company; Duracell, Energizer, Rayovac, etc.
All NiMH batteries have a mAh (milli-amp hours) rating, which is a rating of the capacity of the battery. The higher the number, the longer the batteries will last on one charge. The mAh number you choose, should be 2000 or larger.
I am a struggling, haven't-had-a-paying-gig-yet, almost pro photographer. I can't afford to buy a transistor pack yet for my Canon Speedlite 580EX, but I did buy the Canon Compact Battery Pack (CP-E3)instead. Now I need to put batteries in the compact battery pack!
I need to know which are the best rechargeable batteries for high drain electronics like this flash. Any and all help would be greatly appreciated. Thank you in advance.