[mmastrac]

Supercap or battery: anything with a decent enough density to give you enough time to swap the battery without losing state or being stressed out.

[Cyph0n]

A cap like that might not fit.

[pjc50]

Maybe .. another battery? One with 5-10% the size and capacity of the main one? Although in both cases the power management is a little tricky.

[ZenoArrow]

Capacitors can be made in a variety of shapes, internally they're a simple sandwich-like structure of a couple different types of material:

https://upload.wikimedia.org/wikipedia/commons/thumb/3/35/Pa...

Capacitors are usually cylindrical to increase the available surface area whilst keeping the horizontal size down ( http://www.camion.com.tw/images/en_basic_structure.gif ), but they don't have to be designed this way. Here's an example of a flat supercapacitor:

http://gpsworld.com/cap-xx-launches-0-6mm-thinline-supercapa...