What method of browsing the root filesystem is least likely to make any modifications to it? Hooking to a pc is an option. Jailbreaking is not an option for this question.
Despite the fact Steve Jobs didn't support this idea because it's making things complicated for the users, there are some apps which will allow you to do so.
There is i-FunBox which allows you to browse through your iPhone files on your Mac. And if you're a Windows user, you've more options. There is iPhoneBrowser or iPhone Folders.
All of these solutions will allow you to browse the iPhone file systems using your computer.
If you want to be able to browse on your iPhone yourself, I'm afraid I only know one solution, called iFile. But this is only accessible on a jailbroken device...
I'm using DiskAid which is a good file manager to access the iPhone file system from your desktop Mac or Windows. You can connect to your device through Wi-Fi or USB, access your App contents, Music, SMS etc. Transfers are stable and reliable, for example if you disconnect your USB cable during a transfer the transfer will automatically continue via Wi-Fi.
The root file system is accessible in DiskAid but as the other similar solutions only if your device is jailbroken.