I have opened up several of the portable jumper packs. None of them had diodes on the jumper cables themselves, so they can be recharged by hooking them up to your regular battery and running the engine. They should not be left connected over long periods when the engine is not running.
While they all contain different sizes of 12 volt AGM sealed batteries, the ciggy plugs might or might not be able to flow electricity backwards(possible diodes), meaning you might not be able to just plug a cord with 2 male ciggy plug adapters to charge it. This would not work very well anyway because the limited wire sizes in vehicles limit the charging current anyway.
A fully charged 12 volt battery should actually read 12.8 volts, so a wall wart which only outputs 12.0 volts will not charge the battery. Usually they are 13.7 to 14.4 volts and 600 to 1400 milliamps.
If you have a regular automatic battery charger with a 2 amp setting you can use this hooked up to the alligator clips. A fully discharged 12 amp hour battery would need ~ 6 hours to be brought upto 100%. 12 a/h is the smallest battery I have seen in these jumper packs, with 24 a/h being the biggest. They are sealed AGM batteries and are sensitive to overcharge, so do not use a high amp charger on them, or even a low amp manual charger unless you monitor it closely. Do not let the voltage go up and over 14.8 volts, and when they get warm, they are full. If they get hot, they are being overcharged, will offgas, possibly dangerously, and lose capacity.
The batteries within these jumper packs are designed as starting batteries, versus deep cycle batteries, so they really do not like to be discharged below 80% State of Charge(SOC)very often. The deeper any battery is drawn down, the less cycles it will give. Deep cycle batteries just are more tolerant of this abuse.
While AGM batteries self discharge at a much slower rate than flooded batteries, they will still sulfate and lose capacity quickly if they are left in a discharged state. The length of time they sit discharged and the depth of the discharge dictate how much capacity they have lost. If your jumper pack has sat uncharged for months, It might not be worth the effort of trying to find a wall wart charger for it. You could get one, fully charge it, and then find that when you need to use it, it doesn't have enough juice to jump your vehicle.
I have replaced the AGM batteries inside these units for about half the price of buying a new one. I have also cannabalised them for parts.
When you look at the cost of them you realize you are buying convenience. For the same price you could buy another full size battery that has 5 to 9 times the capacity and use a pair of jumper cables.
. The cylindrical part of the adapter plug that inserts into the unit is 1/4 inch wide with a mid-sized center post and hails from the "wall wart" family tree.