Puck bigger battery

You might be surprised actually - if it's just a joypad with buttons then Puck.js will probably do pretty well (you could make it go to sleep after 15 mins of not being used).

There isn't a guide at the moment, but as a quick run-down:

• The maximum voltage for Puck.js is 3.6v, which is too low to run it directly off a LiPo battery
  
• You could run it off a LiPo with a voltage regulator though (someone must do voltage regulator + LiPo charger combo boards, in fact you could probably change the resistors on a USB power bank's supply)
  
• The simplest method is probably to use a standard Lithium (not ion/polymer) battery like this: http://uk.farnell.com/varta/6127201301/battery-lithium-1-2aa-pcb-3v/dp/463310 - not rechargeable but it's got almost 5x more power than the CR2032. The full AA size ones are even better.
  
• LiFePO4 batteries have the correct voltage range too - and are rechargeable. They could be a really good bet