Stealing directly from ASoIaF, and adding my own twists to suit my campaign, thus far I have:
- Military Advisor (possibly also the captain of the royal guard, but perhaps those are two different roles)
- Political/diplomatic advisor
- Spymaster
- Financial advisor
- Magic/mystic advisor
- Religious advisor
My only objection is that these positions are likely to have Fancy Schmancy titles that go along with them. Also, it might not necessarily be obvious to outsiders what a particular functionary does. For example, a 'Chamberlain' is in theory just someone responsible for cleaning the King's bedroom, but the position has often traditionally extended to mean in practice 'Chief Diplomat', 'Prime Minister', or 'Royal Secretary' - since the real function of the office is determining who can be introduced privately to the King. Certainly the 'Spymaster' is unlikely to be called 'the Spymaster', unless, for whatever reason your culture favors blunt ostentation, in which it might be something like, "Lord X, His Royal Majesty's Master of Spies'. He might however go under a title such as the Lord Referendary, or the position might be so secret that exactly who is the King's spymaster might be something of a mystery (see below).
Some suggestions:
Military Advisor: Lord Marshall
Diplomatic Advisor: Lord Chamberlain
Financial Advisor: Lord Chancellor
Magic and religion are going to be touchy and difficult. The latter most especially because D&D assumes polytheism, so there won't be just one cult vying for the King's ear, nor is the king himself likely to be monotheistic in his practices. Almost certainly as a duty of the office he is called on to officiate at numerous religious observances. So the advisor might as much be an advisor on proper ritual and etiquette as much as a religious advisor. More on that as well.
Are there any positions I'm forgetting?
King's always need a 'Master of Ceremonies', who instructs the King on fulfilling the duties of his office correctly. A King is almost always engaged in some ceremonial duty or the other, and in a magical world the proper fulfillment of these ceremonies may well be a matter of the security and health of the state as a whole. Indeed, in a magical world the most important thing that a king might do is ritual.
King's will need a legal advisor. Although King's are often the source of new law, they are often bound by existing traditions and laws, and called on to settle disputes. Right after ceremony on the list of duties performed by a King is to be the nation's court of appeal, and chief judicial officer. So they almost always have a Lord Judiciary of some sort, who nominally heads the lower courts on the King's behalf, and advises the King on the wording of laws and the existing laws that are enforced in his name.
Beyond having various Great Offices and office holders that he takes advice from and delegates duties too, King's often have a bevy of personal advisors and councilors - usually individuals that are either friends of the King (and are often his cousins or other relations) and can give him council regarding his emotions and judgment without giving offense (and risking being beheaded). The King may often keep a trusted mentor or tutor from his childhood around as long as the old councilor is living, to ask scholarly advice of - "Where is Mercia, and what is their chief export?" for example. And generally, scholarly sorts that can answer questions like that will be on hand and can sometimes achieve high importance. These informal advisors could potentially be more influential than the ones that hold the official offices. Generally speaking if they don't come with titles of their own, the King will grant them something. So Lord Borax, Baron of Ginthia might be according to the peerage a noble of quite small rank, but if he went to school with the King and they've been bosom drinking buddies since their youth, it's quite possible the Baron of Ginthia influences policy more than some great lords in the Kingdom.
If Spymaster is a secret position, it's likely that the Spymaster 'hides' out among these more informal advisors and pretends to be one.
Would you as a player be happy with 5-7 advisors being introduced over one or two session?
Sure, why not? Just make them colorful enough to distinguish between them. If you haven't read it before, track down the old Dragon article on the 'Seven Sentence NPC'. That's a good place to start.