Mountain Lion boot time extremely slow!
This problem may be the result of failing RAM or a failing disk drive or potentially some other hardware problem.

It should really not take more than 5 minutes to boot a mountain lion compatible Mac in the worst case.


Check how much RAM OS X thinks is in the computer Apple Menu > About This Mac. Make sure it is the amount your expecting, for example if your computer when you bought or upgraded it it said it had 4GB of ram make sure its not showing that there is some different smaller number.
Run a Apple Hardware Test for your Mac.
Try running a memory test program, something like memtest
Back up your important data
Try running a disk testing program, something like TechTool Pro 


If all the above fails to uncover a cause try a clean install


Make sure you have a boot-able Mountain Lion Installer, etc or have an internet connection and a "OS X Internet Recovery" compatible Mac
Erase the disk drive, and re-install a clean copy of OS X.
Use the machine for a while, and see if the slow booting problem returns( if there is a problem with the disk the issue should return after indeterminable time). If the problem does not then this would fix it.
If the problem does return I would try a different internal drive to boot the machine with, in order to determine if the problem is with the drive or with the mac.
Repair / replace the failing hardware, note that parts may be covered under Apple's Warranty.