iPhone 4s stuck at apple logo in boot My iPhone 4s 64GB was recently updated to iOS7. 

The battery was completely empty so I had to charge it.
However, the iPhone does not boot anymore and it's stuck at the Apple Logo.

I went to a service center and they say it's on recovery mode and I have to re-install the software, but data will be lost.

I have a lot of important data, how can I save these?
If it doesn't boot and the device isn't jailbroken, you can't get the data off it.

Hopefully you have an iCloud backup that you can restore to after restoring iOS (using recovery/DFU mode).