I'd look for oligarchies in parts of the governments with the following pattern.
Long terms of office - it is hard to gain control if you are being swapped every time there is a new president
Control of subordinates and control of likely replacements/peers.
Take for example the Army. Generals last for longer than a presidency, get to control who goes up ranks and also because there is no outside source of knowledge of the army or military affairs, the possible replacements are only those that they have groomed.
So I would look for that type of pattern. A department I would expect not to form an oligarchy the Centre for Disease Control, where knowledge of infectious diseases and how it can be controlled can be acquired in civilian life and the Director is replaced frequently.
What they want? I'm not sure I can do more than suggest they will want to improve the power and influence of that group. Being a General in the Army is not so impressive if there has been massive cuts, even if your job is safe.
Most long term users on Less Wrong understand the concept of optimization power and how a system can be called intelligent if it can restrict the future in significant ways. Now I believe that in this world, only institutions are close to superintelligence in any significant way.
I believe it is important for us to have atleast some outside idea of which institutions/systems are powerful in today's world so that we can atleast see some outlines of how the increasing optimization power will end up affecting normal people.
So, my question is - what are the present institutions or systems that you would classify as having the maximum optimization power. Please present your thought behind it if you feel you are mentioning some unknown institution. I am presenting my guesses after the break.
Blogospheroid's guess list
Institutions I found significant outside this list is the Singaporean and Abu Dhabi city governments, very rational and increasing their significance in the world, but highly restricted from fooming because of constraints.