It's a simple problem. Sometimes Windows will just halt everything and throws a BSOD. Game over, please reboot to play another game. Or whatever. Annoying but not extremely serious...
What I want is simple. I want to catch the BSOD when it occurs. Why? Just for some additional crash logging. It's okay that the system goes blue but when it happens, I just want to log some additional information or perform one additional action.
Is this even possible? If so, how? And what would be the limitations?
BSOD happens due to an error in the Windows kernel or more commonly in a faulty device driver (that runs in kernel mode). There is very little you can do about it. If it is a driver problem, you can hope the vendor will fix it.
You can configure Windows to a create memory dump upon BSOD which will help you troubleshoot the problem. You can get a pretty good idea about the faulting driver by loading the dump into WinDbg and using the !analyze command.
Knowing which driver is causing the problem will let you look for a new driver, but if that doesn't fix the problem, there is little you can do about it (unless you're very good with a hex editor).
UPDATE: If you want to debug this while it is happening, you need to debug the kernel. A good place to pick up more info is the book Windows Internals by Mark Russinovich. Also, I believe there's a bit of info in the help file for WinDbg and there must be something in the device driver kit as well (but that is beyond my knowledge).