Naming a 4 branched cyclic compound
Your selection of the main ring and the bridges that results in the bridge lengths ‘[9.2.2.2]’ and your counting of the total number of skeletal atoms that results in the name ‘heptadecane’ are correct.

For your proposed name ‘bicyclo[9.2.2.2]heptadecane’, you obviously followed the nomenclature of similar bicyclic systems, such as bicyclo[9.2.2]pentadecane.



This is not a bad idea since the nomenclature of polycyclic hydrocarbon ring systems is actually based on the nomenclature of bicyclic systems.

However, the structure given in the question is not a bicyclic ring system because it includes one additional secondary bridge; therefore, the structure actually is a tricyclic system.



The additional steps for the naming of tricyclic hydrocarbons are described in the current version of Nomenclature of Organic Chemistry – IUPAC Recommendations and Preferred Names 2013 (Blue Book) as follows.


  P-23.2.5 Naming and numbering tricyclic alicyclic hydrocarbons
  
  P-23.2.5.1 Tricyclic hydrocarbons having an independent secondary bridge are named on the basis of a bicyclic system (…). Rings not described by the bicyclic system are defined by citing the number of atoms in the independent secondary bridge as an arabic number. The locants of the two attachment points of the independent secondary bridge to the main ring are cited as a pair of superscript arabic numbers (lower number is cited first) separated by a comma.
  
  The name of the tricyclic system is then constructed by citing:
  
  (a) the prefix ‘tricyclo’, in place of ‘bicyclo’, indicating the presence of three rings in the polyalicyclic system;
  
  (b) numbers indicating the bridge lengths, starting with the two branches of the main ring (…), followed by the main bridge (…), and the secondary bridge (with superscript locants separated by commas indicating its points of attachment to the main ring), all separated by full stops and placed in brackets,
  
  (c) the name of the acyclic hydrocarbon having the same total number of skeletal atoms.


Therefore, the correct name for the structure given in the question is tricyclo[9.2.2.21,11]heptadecane.