Salt rising bread qualifies, though you are unlikely to make it. It is a bit of a process, can be a little fussy, and the taste is somewhat unusual (though I like it a lot). In the words of noted food writer, Harold McGee:

"I've recently come across a fringe fermentation method that, unlike the breads and brews and yogurts and pickles and misos we know and love, isn't run by the usual benign microbes. The engine behind this fermentation method is Clostridium perfringens, a close relative of bacteria that cause botulism, tetanus, and food poisoning. It can eat flesh. It gives gas gangrene its name by causing putrefying flesh wounds that bubble and foam with flammable hydrogen. And it can make something surprisingly delicate and tasty."

https://www.popsci.com/article/science/clostridium-it-can-ki...

There's a classic Australian bread - Damper that's made without Yeast, it just uses self raising flour as a raising agent. https://www.taste.com.au/recipes/damper/80391c40-8eab-4e67-9... Its great to make when camping

I find damper hard to digest compared to normal bread.

I purchased a large package of instant dry yeast and keep it in the fridge, in a mason jar. It lasts essentially indefinitely (I've had it for _years_.)

Part of the reason it's hard to use chemical leaveners in breads is that they only provide so much lift. A yeasted bread can go through multiple proofings and risings.

I know this isn't without yeast, but you can start the development of your own sourdough culture without added yeast.

You can store yeast in the freezer for over a year, so it shouldn't be too much of a burden to buy it.

Soda bread, but just buy some instant yeast and keep it in a jar in the fridge: https://www.amazon.com/dp/B0001CXUHW/

Random thought: Since you're a cake baker, that's probably going to be Cake Flour you have in your cupboard -- lower protein (gluten) content than bread flours, so you might want to get in some Bread Flour instead.