What are the most maintenance-intensive parts of a helicopter?
The question is a bit broad but in general but I will do my best.

First we must look at how maintenance intervals are determined, normally during the initial certification the manufacture testes there aircraft and determines its limitations both in flight, and for maintenance. the regulator generally looks over testing, and documentation and decide if they will approve the 3 parts: the Aircraft, the Flight Manual, and the Maintenance manual. generally the regulator will not do there own tests, they just decide if the testing carried out is sufficient. (this is my best understanding of this, its complicated and a little different depending where/when the aircraft was developed)

Operators who have larger fleets of 1 type of aircraft, or large maintenance/overhaul facility's  will sometimes develop there own maintenance manuals, during these processes they will sometimes demonstrate to the regulator that with newer testing methods etc. that certain components are actually lasting longer than originally proven. Upon approval of there regulator these new documents become law. Occasionally these changes are driven by the manufacture, often when competing for larger contracts ie. Military that wants different maintenance. 

As far as "driving" components most people would think the large items requiring very long overhauls, Ie. rotor blades, transmissions and engines.

To answer your questions on shortest maintenance intervals, older helicopters often have small field serviced items such as vibration damping oil struts, or unsealed bering's that may need to be serviced every day or more. Astars air filters need to be cleaned and oiled something like every 15 hours, and piston engines often need oil filters every 25 hours or less. most of these jobs only take an hour or two.

As far as inspections, all helicopters have a daily inspection that is carried out every day checking for issues.