Is there such a thing as an OLAP cube that can change the number of levels in a hierarchy without having to be rebuilt? The current analytics solution that my team works with cannot handle changes in the structure of one of its hierarchies. If a new level or branch is added, then the entire domain must be first backed up, and then rebuilt with the new hierarchy. 

One of our use cases requires an OLPA model where the domain should be able to gracefully handle changes to the hierarchy structure. Does such a thing even exist?
You will need a parent-child dimension, where each member also keeps the id of its parent; something like

Id; name; parent

1; world;null

2; europe;1

3; north america;1

(...)

123; france; 2

And so on. Whether your particular OLAP solution supports this or not?... that’s a different battle.