I used 18650 battery cells (9) from used laptop batteries and a 75 watt car inverter. The cells cost me about $2 and the inverter was $5. This could charge an i phone from zero battery to full about 12 times.
It would be easy to use a larger inverter and more batteries to power larger devices. With about 10 lb of batteries (100 cells which would cost me about $15 in used laptop batteries) and a 1000 watt inverter you could power a microwave for almost an hour. Many other things like small refrigerators, tv could be powered.
Step 1: Building It
I uses 9 cells, 3 sets in series of 3 cells in parallel. Voltage 12.6-9 from full to empty. Max amperage is about 13.2. I hot glued the cells together and then solder the ends with 22 gauge wire to form the 3p3s pack. I solder larger wires to the positive and negative pack ends.
I then connected the battery pack to my inverter and plugged my soldering iron to make sure it worked. It started melting solder quickly just as if it were plugged into the wall.
I soldered wires to the inverter. I flipped around the part in front of the fuse so that it pushed the spring back to connect the positive. I soldered the negative battery wire to a switch. I soldered the inverter to the wires. I pushed these solder joints through the box for charging and checking the battery voltage.
I hot glued to inverter and charge wires to the box.
Step 2: Charging and Battery Voltage
You can easily connect a multimeter to the wires to check the battery voltage.
You should cut pieces of cardboard out on both sides of the battery pack to check the voltage of each cell. You can the charge cells to balance them.
Step 3: Plugging Stuff In
So far i have tried a 12W soldering iron, an I phone, I pad and laptop.
Just keep in mind the power limits of your battiers and inverter. My inverter can run at 60 watts continuously and the batteries are good for about an hour at this power.