The entire body is conductive and wrapped in a tiny layer of plastic.
And: Fun part: With standard alkalines (thanks Nick) it's the + that wraps all the way round. At the negative you can actually make a small screwdriver glow by jabbing it in the tiny opening between the negative and the body. (THIS IS DANGEROUS! DON'T DO IT! ETC! But you could... :-D )
EDIT: Since you appended your question:
You could look into using conductive fabrics, but these are built up of normal fabric with conductive fibers inside, often metal. These fibers can creep into the gap between poles. You need to make sure of two things:
- Loose fibers can't be allowed to creep inside the gap! (short circuit = heavy damage!)
- The sheath around the battery might be damaged before it is inserted, so you need to protect from that, or creeping won't be the problem, it will be full-out contact through your wrapped fabric.
For the creeping problem you may be able to find plastic rings they use in building NiMH or NiCd battery packs to protect terminals from the wires running across. If you are going to be Alkaline, Zinc-Carbon (again, thanks Nick), NiMH, NiCd and Whatever chemistry compatible you need to protect on both ends.
To protect against sheath damage you need some non-conductive fabric or material otherwise in between, that actually wraps around, or this will be a problem at one time and you are not going to enjoy that.
End of EDIT
This is also something Dave Jones highlighted when he was tearing down these "battery booster clips". Next to the facts that many of the claims are exaggerated and/or bullshit, the entire metal clip is connected to the negative, which could cut into the plastic and short out your battery, which is a safety hazard.
That is beside the point of your question, but it's a fun little distraction when thinking about AA batteries, to have Dave explain all their properties to you:
Dave Jones about AA battery booster clip, if you're interested.