Depending on the intensions of the players and the Game Master, the encounter can be comprised of a simple exchange of words, or blow into a full-scale combat scenario - swords/guns/plasma rifles blazing and all.
One of the toughest parts (at least in my experience) of running these encounters is "playing the king". Namely, playing that powerful individual the characters are interacting with. Think about it: in most role-playing games, the heroes (the player-characters) are powerful individuals, most often more resourceful and, well, important, than the high-ranking non-player character they are interacting with.
Classic D&D adventures (such as the well-known King's Festival) have the player-characters act as heroes, otherwise someone suffers (in this case, the town's people due to the possible cancellation of the festival).
So when I arrange for the player-characters to meet the "King", I often see a much different approach to the encounter than that taken by my kids (or by a younger me as a player).
For me as a young D&D player, meeting the king was meeting someone so much powerful than my character, wealthy and worth of my highest respect. But as a more mature gamer, you treat that same king somewhat differently, since there is always the question of sheer power.
Statistically speaking, if a mid-level, well-armed group of adventurers decides to act funny and draw steel in the king's court, there is a good chance (unless the Game Master stomps the so-called heroes with his god-like foot) that the king ends up with a blade at his royal throat. This is especially true in D&D, where a well-placed spell or a sneak-attack can end a battle before it began.
Obviously, the Game Master can wave his hands and make this problem go away by making the king more powerful than the heroes (a difficult task in D&D 4th Edition, see Epic Characters for more details), but such an approach is problematic in my opinion. If all the "powerful" individuals dealing with the player-characters are more powerful mechanically (rule-wise) than the heroes, why do they need the heroes?
The opposite approach - to accept things as they are and let the heroes be more powerful (in terms of resourcefulness, wealth, magical power or combat prowess) - can lead to players feeling that these powerful individuals they are dealing with are buffoons.
The way to solve these issues came from an unexpected direction: Chess. My son was recently taught the rules of the game by a relative, and I had to learn them myself so we can play together.
In chess, every tool have a specific mode of movement across the board. The goal of the game is to corner the king in a way that every movement ends with the king being captured. I totally expected the king to be the "best piece on the board", but it turns out that it's not.
The king, while being the most important piece, is usually the weakest piece until a later phase (the endgame). All other pieces are used to guard the king, and manoeuvre to capture the opposite side's king.
It occurred to me that in D&D, kings should be treated the same. Mechanically, kings should not be the most powerful characters around. Mechanically, even a low level player-character might be a threat to a king. But the king should be surrounded with people (non-player or even player characters) that need his safety guaranteed. As in Chess, the king should seek safety behind friendly pawns, taking an active role only to help other "pieces" achieve their (and his) goals.
This way, a king should have the protection of wizards, knights, priests, merchants, diplomats, assassin guilds, inquisitors and even loyal servants and cooks (after all, the kings eats what the cooks prepares). There should be layers upon layers of protection, some obvious (such as the armoured knights standing beside the throne) and some subtle (such as the trained assassin posing as a manservant, or the invisible wizard lurking around when the king accepts guests).
Even if there are no "mechanical" layers of protection in the form of mechanically powerful individuals guarding the king, there should be contextual layers of protection such as player-characters (or non-player characters) with something to lose if the king dies/loses face, or whole organizations built on top the king's influence - organizations that will not stop until the individuals responsible for the loss of their source of power are caught and taught a lesson.
While it's possible to peel all these layers and "capture" the king - the way to do so should not be easy, unless someone makes a dire mistake. This keeps things fair, and can help a DM to give a plausible explanation for an unlikely event. But more often than not, the players need to feel that the meeting with such a "king" is a special event, full of intrigue, opportunity, and danger - some obvious, some not...
"Welcome, adventurers." Said the King. "Word of your deeds reached my ears, and I would require your services in investigating a delicate matter - the recent assassination of my court wizard."
"Surly your majesty have resources far beyond ours," Smiled Aladon, looking back at this group of fellow adventurers. "Why would you require our services, which are, ahm...so expensive?"
"My resources are indeed far beyond your own." replied the king with a smile of his own. "Using my resources, I found out that the Laughing Death Cabal is involved. Too dirty of a business for a diplomat like me, but hopefully not below your standards, yes?"
Aladon cursed under his breath, but kept smiling. His fellow adventurers gulped. Gold seemed a lowly prize for dealing with the Cabal and their flesh-eating assassins...but helping the king in this might prove fruitful, and backing off now will make Aladon and his fellow adventurers the joke of the court...
Damn the lord who arranged this meeting, thought Aladon. Reward worthy of kings, the lord said, but Aladon felt that this whole charade was orchestrated - leading them all straight into a trap. But why, and most importantly, who...?
photo credit: loco's photos via photopin cc