Can the Skilling Backpack regain charges?

Solution 1:

It looks like it was once rechargeable, but you no longer can charge it unless you already have some Powerpacks. Powerpacks were available from Treasure Hunter when they reintroduced the skilling backpack for a limited amount of time (between August 23, 2018 and August 27, 2018) during the Return of the Pack promotion.

Note that the original skilling backpack was from an earlier event in July 2016 (the Boat to the Arc event). It had slightly different mechanics on how it worked, but during the Return of the Pack event everyone was able to obtain the backpack from Treasure Hunter, regardless if they participated in the Boat to the Arc event. If you had the original backpack from the Boat to the Arc event, it was replaced with the new one from the Return of the Pack event upon opening Treasure Hunter. Should another promotion like this occurs, they will likely do the same thing and either replace the backpack you have now, or give you a new one completely if you don't have one to begin with.

Unless they add Powerpacks back into the game, it looks like you cannot recharge it.