Iphone 3g problem with just a apple sign? 100% battery Not charging after going dead!

The battery is probably dead. You can locate a repair service who will replace it for you for under $50, or you could do it yourself, it's not very difficult. You can easily find detailed instructions on the web, and the replacement battery costs about $5 on ebay.I found good instructions here, that will give you an idea of whether you want to undertake the repair yourself.
http://www.imore.com/weekly-mod-replace-battery-iphone-3g3gs

Unfortunately there is not a lot of support from Apple or the jailbreak community for iphone 3g, so there isn't really anything you can do to make it operational again.

Most likely your iPhone battery is dead. If you have the AppleCare and hasn't expired Apple will replace your iPhone battery free of charge.

If it has expired you can pay about $50 (I think) and Apple will replace the battery. If you think that's too much go to Amazon/Ebay and look for an "iPhone Battery Replacement Kit." Remember if you do decide to do the battery replacement yourself. You're voiding your Apple warranty & they won't replace it.

just press the sleep button and home button at the same time until you see the small apple icon and then try charging it again.

http://support.apple.com/kb/HT1808

Let it charge for a few hours

Then try pressing the home button and the power together for about 10 secs

It might then restart, if not then the battery could be dead,

have you got alot of applications running or installed? turning your 3g and bluetooth off will save battery while on the move. try letting the battery run dead then charge constantly for 24hours

Apple will be able to help, they can find a way without restoring your entire phone, you can call apple at 1-800-MY-APPLE.

Your battery is deteriorating. Send it in to apple to get it replaced-
$40- 50. when back, fully charge and discharge 5-10 times, and do NOt leave it constantly plugged in.

Try an a/c charger, many times the usb ports don;t put out enough to give it a good charge.
If it's a 3G you have to have a charger that works on it.