Upon reading this article:
http://www.gizmag.com/tin-whisker-battery-anode/22905/

One of the first thoughts which struck me was “a conductive aerogel (e.g. carbon aerogel) would likely extend life even further”. Aerogels ordinarily have very high electrical resistance (because they’re mostly air), but certain types can be made to conduct, and they seem like the perfect material for creating compact supercapacitors.