You normally only have to claim your baggage and then re-check it when you first arrive in your destination country. This allows you to clear customs, which you are not be subject to when arriving on a domestic flight. Therefore:
London - Chicago - Los Angelas > You claim your bags in Chicago to clear US Customs.
but...
Los Angelas - Chicago - London > You claim your bags in London only.
and...
Cairo - London - Los Angelas > You claim your bags in Los Angelas only.