Mountains are higher than the atmosphere? Is it theoretically possible that planets exist with mountains so high that their peaks overtop the planets atmosphere? And which physical laws are relevant for this question? I'm just curious.
The question came while I imagined the atmosphere like a second ocean above our ocean of water. And I thought it would be nice if such a gaseous ocean could have islands as well.
This is a bit of a gray area, as an atmosphere doesn't have a clear boundary. That being said, Olympus Mons on Mars is so tall, the atmospheric pressure on top of it is only 12% the average pressure on the surface of Mars. That's near vacuum by terrestrial standards.

https://en.wikipedia.org/wiki/Olympus_Mons#Description

In general, for this to happen you need:


a pretty thin atmosphere to begin with
some exceptional geology that ends up producing very tall anomalies like Olympus


It's not a very likely combination, but it can happen, as seen on Mars.