It appears that I have been far too broad and general in my comments - guilty as charged.
No worries, its just a discussion.
Now, to take one valid rebuttal and address it. If one player at a table has an archduke for a PC, I would expect (perhaps inaccurately) that A. this is not the result of a fully random chargen, but rather a deliberate campaign setup unanimously agreed to by the ref and players alike* and B. the other PCs are senior members of the archducal retinue, household, huscarles, etc. - the kind of people that do accompany an archduke on the move. The key for play dynamics is that their relationship with the archduke is such that he views them as confidants and advisers, not lackeys to be ordered about; the latter such is the purview of NPCs.
That would be one way to do it. Another would be to allow each player play a diferent Duke or Archduke, each with their own retinue, goals, motivations, etc. It could be done like the default model in Ars Magica where each player creates a mage and then a retinue of other characters which are generally played by other players. Each game session focuses on the storyline of each mage, rotating from one to the next. So sesion 01 might be about Archduke Alpha, with the other players playing members of his family and staff. Session switches to Archduke Beta, and so on. In some sessions you might have two or more Archdukes active in the same session when several are interacting in some way.
As to what kind of adventures they would have - that depends on your view of the 3I, what kinds of non-routine things might happen to an archduke and retinue, and the kinds of stories that ref and players alike want to tell through their gaming. Scarcity stories are too forced in such a situation, except as a single, non-recurring plot line (marooned or the like) or a major change in the setting and the direction of the game (e.g. major political upheaval results in the archduke losing their title). Sufficient invocation of the Rule of Drama (and as necessary, the Rule of Cool) can suspend disbelief as necessary.
Typically I find the following things are key plot elements in this style of play:
politics - intrigue, rivalry, espionage, alliances, political philosophy, etc.
social - relationships with friends and family, attitude and relationship with people you have authority over, social problems (unrest, crime, economy, etc. and finding solutions to these).
personal - personal motivations, goals and beliefs become central plot devices. Dealing with personal demons and how that affects others.
I also typically find the following things become uncommon:
Combat - not much personal fighting... maybe a duel but even this should be a dramatic event and its more about the story that led to the duel with the actual duel simply being the climatic result. Occasionally if there is a war there might be a chance for some heroics there. Not many bar room brawls or the like.
Exploring - not much in the way charting new territory, crawling through ruins or dark caverns or what have you... you hire people to do this, then react to what they find.
Blue collar stuff - small time (struggling) merchants, smugglers, etc. don't often enter the picture except as NPCs, the PCs won't be doing this sort of thing... again, they hire people to do it.
At this level the PC is the Patron, the GM should not be leading them into adventure but instead stepping back and letting the PCs set things in motion, pursue their own agendas, etc. The GM is there to run the universe they exist in, but shouldn't need to "prompt" the players with "so you're at the local pub on the planet you just visited when a Patron approaches you..." The PC should be the ones doing the approaching, they are the ones now responsible for setting events in motion.
At most the GM perhaps throws out a major event they all then react to. For example, say you have five players and each is playing an Archduke. You as GM decide to throw a war at them... the K'kree just launched an attack. This will directly affect some more than others, not all of them will be directly affected, while others will see worlds under their rule become battle grounds. Story arcs could deal with how each reacts to the war in various ways, both as a governing body and personally. As an archduke how do the prepare, defend, help those attacked, counter attack, negotiate peace, and so forth? On a more personal level how do they react when for example they send the 453rd Antarran Fulsiers into battle, in which the Archdukes nephew is an officer... reports come back that the battle didn't go well, the Fulsiers took heavy losses and there is no word yet regarding the nephew who is ultimately listed as MIA. The PC can't go charging to the rescue personally, as an Archduke they are both very powerful and at the same time in ways helpless, how do the PCs react to that? What do the do if the nephew was captured? If the enemy uses them to bargain with? Do they sacrifice their people to protect the nephew? Do the needs of the many really out weigh the needs of the one? What about the aftermath of their choice?
At this level of play there is generally a lot less dice rolling I find. The focus is on the story, not on whether you managed to shoot the gun out of the other guy's hand. Do NOT let it become just a game of numbers and bookkeeping. The story is in how any why they make choices, and how they deal with the consequences of those choices. For example, social problems cannot be solved just by throwing X amount of RUs at the problem... make them work through it, and sometimes face the reality that there are no perfect solutions, or that sometimes the only real solution is one the public will not accept, and that generally, no matter what they do somebody will be unhappy about it and that in itself become another problem. You can draw on real world history and current events for a wealth of plot ideas... many of them repeat through history.
It makes for a very different style of game but it can also be very enjoyable. Not necessarily better (I hate those "this is better" arguments, personally I enjoy the variety of options... struggling free trader last campaign, an archduke this campaign, next campaign a merc captain and the GM decides just for irony it will be set in the war my archduke fought, same war from a different perspective) just different.
Think of it as being like a camera focus. The "struggling free trader" style campaign has a tight "camera focus". The larger events of the universe are "out of focus", simply there in the background but not generally significant to the story. The "scope" is limited to the free trader's perspective. With Archdukes the camera "pans out" to show the wider "scope" of the galaxy. Now its the free trader who is "out of focus", and the larger events of the universe become the main story, the perspective has switched to that of the Archduke.
As I mentioned above, it can also be fun to start with some Archdukes, run campaign (as short or as long as is fun) and gaming out a couple of decades of game time. Decide what major events will occur, then let them react to them. Take notes on everything.
Next campaign, let them play free traders or mercs or soldiers who now live in that same time period they previously gamed out. Same events with the archdukes doing the same things they did, but now they get to experience it from a different perspective.
One final note, the above styles of gaming requires players who enjoy story telling. If your group mainly likes shooting things, it probably won't work for them. As always, its important to know your group's gaming style.
