How to refract a while loop with a necessary initial setting?
I'm having trouble figuring out exactly what your code is doing. It's called get_exception_region but it appears to do a lot of printing. Is that just for debugging purposes? I'm going to assume that's the case, and thus that get_exception_region returns the row (line number?), text, and region of the line following the exception. Here are my suggestions, given that assumption:


Change the docstring to reflect what get_exception_region does, not how it does it. From another angle, describe why someone would choose to call get_exception_region, not what code they would have to write in its place. If there's any code in the docstring, it should show how to call the function, not reminders on functions it may call to implement itself. If you need the latter, comments are more appropriate for that.
If you have control over it, the name except_region.a doesn't convey much. Consider using a more descriptive name than a.
As mentioned by @ChrisWue, drop the True and from your while condition. At best it does nothing; at worst it distracts readers and slows the program.
Consider if rewriting your while loop as a for loop (for row in range(start_row, start_row + max_lines_in_exception): ...) would add or diminish clarity. This would avoid code with a slight smell (row = row + 1), but in this case I'm uncertain whether the replacement would be better.
Avoid catching Exception; if you must suppress most exceptions, it's typically better to catch BaseException (at least in python 3). That said, since this code quickly re-raises the exception, this advice is less crucial than usual.
Avoid accessing attributes like tb_frame, f_code, and co_filename in a function not specifically geared towards this. Instead for the usage in your exception handler, prefer using tools from traceback such as print_exc; in other cases inspect can be similarly useful.
Remove dead code. Having raise e followed by sys.exit(e) is misleading, since the raise will unconditionally prevent the exit from executing. Similarly the pass after return will never be reached.
If all you are really doing in this handler is printing out a partial trace, consider omitting the try/except handler entirely, and letting the caller be fully responsible.