Any way to recover data from a semi-bricked iPhone?

My iPhone turns on, shows the white apple logo, and then after a few minutes shuts off. It is not a battery issue, it happens even when it's plugged in.

Is there any way I can get the data off the phone without resorting to a hard-drive recovery company?

Maybe some kind of program that can access the hard drive through the USB port? Or maybe I can open up the phone and somehow get the data off the hard drive?

I know that I can restore the phone through iTunes and have it working again. The problem is that I didn't back up the data so I have 400 or so notes saved only on the phone. If I restore it, they will be gone forever.

Just in case you are wondering, I haven't jailbroken or even attempted to jailbreak the phone. This problem just happened randomly while I was using the phone. It was frozen one day, I tried a hard reset, and ever since then I can never get it to work. I read that some people got the same issue via a remote wipe, however, I'm not subscribed to mobile me, so there is no way that could have happened.

If there is an Apple store in proximity, book an appointment with their tech. If PhoneView does not recognize the phone, I believe this may be your best chance to get the data out.

Here's a late answer for those following in your footsteps (like me).

Huzzah! Editing my previously negative answer to report success! SSH in and get YOUR data off YOUR phone, thanks to a great tool that's so new it doesn't have a cool name yet: shh_rd. The author's blog post has a video walk-through.

If ssh_rd wasn't available, then Apple might be your best bet if your bricked iPhone is under warranty. If you have a 3GS or newer then a forensics specialist is probably necessary. Details below ...

There are very few tools that can access iOS device data directly - and these are usually available only to law enforcement and forensics agencies. (Though there is some interesting work being done on Linux with a library called libimobiledevice.)

One is iOS Forensic Research maintained by Jonathan Zdziarski who wrote the (now dated) book on the subject of "iPhone Forensics". Another is the Elcomsoft iOS Forensic Toolkit. There are also hardware assisted solutions like Cellebrite UFED.

Assuming you don't have access to these kinds of tools, the last resort is to restore the iPhone using iTunes, then jailbreak it and dump the filesystem. The 3GS and newer models use hardware to encrypt their filesystems so recovery is probably not feasible.

Filesystem dumps from phones prior to the 3GS can be scanned for files using software like PhotoRec.