I've had bugs that took weeks or months before a solution was found, but eventually all bugs do get fixed. Aside from the classical non-debugger bug-tracking techniques like disabling parts of the system until you get a minimal test case, I've used these techniques:
Looking for better debugging tools. A new perspective goes a long way. Xdebug is something I started using in PHP only because of a performance bug that I wasn't making headway on.
Studying the technology that the bug is located in. This has helped to debug an outlook add-in. It had random errors that made no sense and that google searches turned up zilch about. By researching outlook add-in best practices, COM and MAPI programming, we got a clearer picture of what could go wrong, and thought of new things to try to fix the bugs, which eventually did fix them.
Trying to exacerbate the problem. If there's an issue that only happens occasionally, I'll try to find ways to make it happen constantly. This has helped to track down errors in web apps under IE and also to narrow down a crashing bug in the flash plugin.
When all else failed, I've rewritten the subsystem that caused problems from scratch. This may take a few days, or even weeks, but if you're stuck on a bug, and can't resolve it, and customers won't take no for an answer, what else can you do? This doesn't always fix things, but if it doesn't, you usually get a clearer picture of what's going wrong.
I've noticed a few commonalities in these bugs that I get stuck on for weeks:
Asking 3rd parties for help rarely helps, and it's generally not a good idea to wait for someone else to come save the day.
Almost always the fault is inside some 3rd party closed source technology, especially when using obscure parts. IE had nasty bugs when trying to use client certificates. Flash didn't deal well with randomly generated drawing instructions (some of which were nonsensical). Outlook doesn't like it when you try to change form layout dynamically from code. These days I've learned to respect the "comfort zones" of proprietary tech.
