Closest to it via mindless searching on electronics message boards + mouser.com is the VL1220-1HF, which is indeed a rechargable cell. At a capacitance of only 7 mAh, who knows how long this really lasts. The only way to actually test this is to not use the G6 for an extended period of time. The hole in that test is that once you actually turn it on again to see if it lost the save or not, you are effectively recharging.
My original statement is indeed wrong based on Rykar's "bullshit" evidence he so graciously provided, but I still believe my second to be correct. There is no need to go out of one's way to try to charge the battery.