If you think about temperature, and what it does to combustibles, I think even holding a 300° brick to a plastic gasoline can wouldn't do much. Maybe melt through the can, and the gas would evaporate super-fast into a pure-white foggy mist ... without ever igniting. How do I know this? Well, I've blown many a fuel line in my day, and have had 700° steel sprayed with gasoline without an issue. Also have seen a 300° Harley-Davidson cylinder head (while sitting in traffic) become doused with about 2 gallons of gas - again without incident.
Now think about it - if your chimney got to be 300° on the outside, just imagine how hot it would have to be on the inside! My guess would be 4 to 5 TIMES that. So if gasoline won't combust (without a spark ...) at 300°, I can place a very sure bet that wood won't in a million years.
I've got a fairly decrepit, unlined and crumbling masonry chimney that also runs through the center of my house. It also jogs in 2 different directions on it's way to the roof. There's plenty of low to zero clearance spots as well. While burning wood, I've had 4-500° stack temps result in a chimney temp of maybe 150° - uncomfortable to hold your hand on, but no possible way it will ignite much of anything. Been burning coal and oil in this chimney for 10 years. Cleaned it for the first and only time in spring of '12. It's still good.
This place is 150 years old and an absolute tinderbox. Still, I'm 200% confident that this chimney will never cause a fire. I did all my worrying and testing in '05 when I got my first stove. All my worry was unfounded, brought upon by media stories of horror. It's good to be safe, but unhealthy to be unreasonable.