Ugh. The 'no password, no work' thing is a recent Apple development. I ran into it on my system and ended up having to do a clean install on a partition I fully erased.
I suppose, if you have the ability to, go online and look up how to burn a bootable DVD image. Download the 10.8.4 image then use the Disk Utility to make a boot image of it. You'll need a dual-layer DVD or a USB drive (or memory stick at least 8GB) If you've got an optical drive, burn the disk, if you've a USB device, use that.
Then get a pocket drive and copy your important files off the system or let Time Machine do a full backup.
When the backup is done, insert the disk and hold down the C key until the machine boots off of the DVD. At the point you get to the start window, go to the menu and choose the Disk Utility.
In the Disk Utility, reformat the drive. You go to the actual DRIVE (not the Macintosh HD, but the root above it that is the actual hard drive device)
I always do this and *partition* it into several volumes, the benefit being, you can store all of your music, documents and importans stuff on that secondary storage volume you make - it will have it's own index and then the first volume is where you put the OS and it will have it's own index as well. That way, if the OS throws a wobbler and you need to reinstall, you don't lose your data.
Depending on the size of your hard drive, it is good to give your OS partition at least 60GB and optimally 100GB, and the rest for storage.
Name your volumes, use the little box sliders to manually size them or put that number in the size window, select for the format, Mac OS Extended (Journaled) and hit Apply and then the Disk Utility will quickly create your new partitions.
Once finished, quit the Utility and it will return you to the OS installer. Select the volume you want to install to and it should start up and install.
Once the installation is done, you'll get the new machine startup then you MUST put in a password for the computer. Once you're in, and to the desktop, don't bother with any Mountain Lion updates, go right to the App Store and get the Mavericks update and let it install the app to your Applicaitons folder, then do that install.
On a side note, there MAY be a way to make a disk image from the Mavericks application and avoid the entire Mountain Lion step.. I honestly do not know, but a bit of hitting the search engines may find out if there is. I think that if you looked for "make Mavericks boot disk or boot image' you may find something.
Good luck!
