Maximum angle depends on the type of rock and on the type of your boots.
There are too many factors to think about, so maybe the question about maximal angle is not useful. Consider the following:
- Is the rock wet or iced? If yes, this may severely limit your walking abilities.
- Is the rock covered with a thin layer of sand or gravel? (you will slip all the time)
- Is the rock crumbling, with a potentially dangerous fall?
- Is the rock a monolithic limestone with small sharp ridges caused by rain, like this?
Just found an example GPS log of a hike, where I went up and down such slopes with average angle of 15-40° (average 30°). I had light flexible boots, which, I imagine, improved traction - so this may be close to the most favorable combination.