How to deal with a chain of several linked 1:n:n*m:n*m*k "join" explosion?
It sounds like you're talking about a self-referential relationship, especially with the inclusion of "manager" titles. This is a common pattern, especially with employee tables. While they may implement it differently the main RDBMS vendors typically offer recursive capabilities. This means that with a special syntax you can easily query this kind of a relationship without writing the hundreds of JOINs you would otherwise need. An example for Microsoft T-SQL recursive capabilities is here and focuses on manager/employee relationships.

Every row should have a ManagerID or Parent attribute, and this can be NULL or the Employee's own ID in the case of the leader (e.g. CEO). This allows you to correctly identify your root node. If you need to traverse down from different levels simply change the start point for the recursion.