If I want to register or renew www.therandomcatsite.com, I know for sure that I can pay GoDaddy about $12 to do so, and GoDaddy notes that the ICANN fee is $0.18. Is it possible to personally file a registration or renewal with ICANN (or whatever central organization handles registrations) and only pay the $0.18 fee?
The short answer is: no, you can't.
The long answer is: yes, you can. But in order to do so you'll need to be accredited by ICANN. See here.
ICANN doesn't provide domain registration services. They oversee the functioning of DNS and various other high-level things, including accrediting registrars. (FAQ here; especially see "What is ICANN?") Providing registation would probably be a a conflict of interest, actually.