One of the things I really like about Classic D&D – especially the Frank Mentzter BECMI/RC version – is the treatment of the post-Name Level game. That is, how the emphasis changes, and the way that's encouraged by the game mechanics. This is most evident in the character options available in "downtime" between traditional adventures.
At Name Level, Magic-Users get to conduct magical research. This is a much bigger deal than it might at first appear. One reason for this is the content of the research: M-Us can craft magical items. Crafting magical items is a very powerful tool, as most veterans of 3.x D&D are well aware – if only because the ability to easily make scrolls of uncommonly used spells and wands of commonly used spells makes the M-Us spells-per-day limitation much less limiting. What's significant about crafting magic items in Basic is that, unlike in 3.x, there is no prerequisite in feats (or anything else) to craft, nor is there any experience point cost; just money and time (and M-Us of this level are bound to have a good amount of money). On the other hand, there is no exhaustive shopping-list of items and costs, which puts things more in the DM's hands to decide what can be made, what can't be made, and what requires a quest for an exotic component, which helps keep it all under control. Unlike AD&D, however, the tone isn't all about discouraging crafting magical items in favour of gaining magical items solely as loot. So the magical research rules in Basic are powerful, but still give DMs the last word in what's possible.
Other classes have options too. The Expert Set allows PCs to build strongholds, claim lands and raise private armies, but the Companion Set opens things up much wider by adding dominions and their management. This is a sophisticated minigame that can play out like an empire-building computer or board game, but it feeds into the core D&D adventuring game too. In a beautifully simple extension of the XP-for-gold reward system for adventuring, dominion income is also worth experience.
Fighters are the quintessential land-owning high-level characters. They gain automatic followers if they build a keep, plus are likely to be offered land in a feudal arrangement. Should they choose this option, the additional XP that they gain from this between adventures contributes to their level gain, which goes a long way to topping up their THAC0 and impressive higher-level saving throws to keep them strong and viable when the Magic-Users are coming into their own. In fact, the usefulness of domain management to the Fighter is such that in a way, the special abilities of the well known wandering-Fighter options of BECMI (Paladin, Knight, or Avenger) are compensation for giving that up.
Clerics are more of a middle ground. They probably get financial aid from their religious hierarchy in building a stronghold and free followers as well, so they are also well suited to the domain-management minigame. On the other hand, they can take advantage from magical research (although not to the same degree as Magic-Users), so they don't get a subclass option comparable to the landless Fighters.* It makes for an interesting choice – be landless and spend downtime making items to get more "juice" during adventures (especially for healing), or run a dominion for the XP to go faster through the upper levels and get to the Righteous Firepower high-level Cleric spells.
Thieves are a bit of an odd one out (as they often tend to be in BECMI). They build hideouts instead of strongholds and get apprentice Thieves instead of soldiers, with a place in the Thieves Guild hierarchy. Actually running organised crime is left to the DM, which feels weak compared to the level of detail given to dominion management. Wandering Thieves get a tantalising hook in the form of a chance of a treasure map or similar per game week, but nothing more systematic. Crime really doesn't pay in BECMI, which is unfortunate for the Thieves. Should the DM flesh things out for the guild thieves, though, there's lots of potential for gangster goings-on with lashings of experience points for ill-gotten gold.
Even Magic-Users can build strongholds (a tower in their case), but it's a more casual affair; they get a few apprentices, but don't really run dominions. One fun quirk is that they can build open dungeons beneath their tower, which draw monsters to lair there. By and large, though, M-Us have more important things to do than manage people.
Demihumans have special rules for strongholds, which – like the racial classes themselves – helps to emphasise that they're not just like humans. For instance, elves build strongholds that blend into the forest, and gain the friendship and aid of the woodland animals. They don't get any special subclasses for forfeiting a stronghold, which makes sense considering they're at or near their level cap anyway.
These options help to change the texture of the game beyond Name Level, and in a gradually increasing way. At higher and higher levels, magical researchers get more and more resources for ever greater projects, while the dominion rulers' territories grow to grander scales with even greater experience rewards. So – while it's never actually forced – incentives are there to spend increasing amounts of in-character time doing non-adventuring things. This gives higher-level gaming a very different style to lower-level gaming (not just more of the same with bigger numbers), and also emphasises the unique style of each of the character classes.
* Neutral Clerics can become Druids, but I think that the trade-off of armour versus the extra spell list is an evenly matched one, so it doesn't belong to the same category as the Fighter options.