Theoretically then:
You can have a vertical opening for any two stories, with or without stairs. Agreed
You can have a vertical opening for stairs up to four stories with size limitation, sprinklers, and draft stops. Agreed
Not codified as such but probably you could go at least four, maybe more, if enclosed in a "shaft" composed of opening protectives, theoretically, if you can afford it. If it's enclosed in a shaft with opening protectives it's an interior exit stair, not sure what you're saying here...
I don't think the fact they are exit access stairs, required or not required, is a factor. Disagree. If an EAS is "required" it is serving as a path of egress travel to an exit. I tried to illustrate my point on the attached.