Try Volume Down/Home/Power (from when it's turned off). If you get into the download screen (there should be a warning and press volume up for this, press volume down for that - I forget the exact wording), you can flash a new ROM which, unless there's a hardware problem, will get it running again. Post here again if you get that far and we'll go further. (Contrary to any warning, you CAN pull the battery or turn the phone off at that point without causing any problem.)
If you can't get to download mode, you have 2 options:
1) Buy a Samsung Dongle. It'll take a while - most of the ones you'll find on eBay come from China, which means 2 weeks normally, but at this time of year, maybe more. They're about $2 including shipping, and that's a ripoff - there's about 25 cents worth of parts and less than a dollar to send the envelope to you. (If you had a microUSB plug and a resistor [of the correct value], you could make one in about 10 seconds plus the time it took to get the soldering iron up to temperature.)
Once you get the Dongle (there should be instructions, or search samsung download dongle), you can force the phone into download mode unless the hardware is blown. (They're used to revive totally dead Samsung phones - ones that act as if there's no battery in them even when there is - due to things like all the firmware having been wiped.)
2) You can bring the phone in for repair. At worst you need a new motherboard, and that would probably cost a bit less than a new S3. Just reflashing a ROM would be a lot cheaper. (Even the people at a Samsung Experience desk at a Best Buy might do that for you - try. All it costs is time and a little gas.) Or you can use this as an excuse to get a newer fancier phone.
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