Personally I handle "the math" of domains in all of my campaigns and simply presented my players with the inputs and outputs they control. (And I had similar complaints from my players about building their 3.5 characters, which also took a spreadsheet in their opinion, and adopted a similar solution.) - Alexander Macris, ACKS AuthorThis is the sort of advice that should go in the DM's section of the book :(
This is also problematic solution, however, because when I am DMing ACKS I already have entirely too much stuff on my plate. I could automate domains, sure, but that's just spreadsheeting taken to the next level.
I guess the lazier solution would be to just nullify domain growth over the timescales most of my games last. You want a better domain? Go and take it. I'm a little dubious that you could reasonably civilize an uninhabited patch of wilderness in five years in mythoclassical conditions anyway. Civilization takes lifetimes, conquest takes a season, and pillaging away lifetimes of civilization takes a weekend. Nixing domain growth would provide a player-comprehensible reason for civilized humans to go to war with each other, rather than just building up to prosperity.
Which brings us nicely to today's topic. Previously I have considered growth rates derived solely from reinvested income; this neglects the players' primary source of income, namely adventuring.
To get a fighter from 1st to 9th level takes 250kXP. By ACKS' 80% rule, this means he will have earned in total around 200kGP. Some of this will have been lost as resurrections, reserve XP, and henchman wages and such, but let's assume a fiscally responsible party for the sake of easy math. During this time, each of the other members of the (stereotypically) 4-man party has also earned 250kXP, so the party's total assets are around 800kGP. An enormous citadel sufficient to secure a 24-mile hex of wilderness costs 480kGP (30kGP per 6-mile hex * 16 hexes), and direct agricultural investments to bring the population up to borderlands cost around 270kGP, for a total of 750kGP.
So hypothetically a near-domain-level party who was really good at saving money could just show up and plunk down a castle and a bunch of irrigation ditches and manage two years of domain growth in a season, but it would take almost all of the money they accumulated during their entire careers.
There is, however, the question of "how long does it take to actually amass 200kGP and hit name level?" Exponential growth is a pain when it's in your way. Previously I postulated that the path to levelling was expeditionary wilderness play, where you follow treasure maps to high-value treasures, but I'm reconsidering that. Treasure map max value is something like 30kGP (plus a couple magic items; more on those shortly). When you're dividing things four or more ways, that's doesn't go very far; you need 25 of them to amass 750kGP, and most maps won't be that valuable (you can do better with Treasurehunting hijink maps since those scale as a function of your thief's level, but they're still not going to pass 30kGP on average until 9th level. The thief does hit 9th earlier than anyone else though, and the treasurehunting map value is pretty swingy so you could follow up on high rolls and ignore low rolls).
Consider also the dragon-hunting strategy. A mid-level party can probably deal with adult dragons, who have treasure type Q+N. Q+N is in expectation about 31kGP, so this is somewhat more efficient than pursuing treasure maps, though also riskier because dragons are guaranteed to be involved, and ACKS' dragonbreath pretty much vaporizes people. You're still looking at 25 adventures to get from 4th or 5th to 9th, though, if you can even find a sufficient supply of dragons (or other high-yield but killable monsters).
Ultimately this is not unreasonable but the problem with wilderness expeditions is that there is typically substantial logistics and travel involved, maybe a random encounter or two, a big fight, treasure, and then more travel and logistics and random encounters. One expedition per session is not unusual in our experience; at one session per week, this puts the mid-levels at around six months of realtime. This roughly matches our experience; Tim's fighter started at 1st and was around 7th after six months of play, and still had a good way to go to get to 9th.
... you know, I should just go full Monty Haul and double treasure in future ACKS games. Treasure is good for levelling rate and good for player morale. Nothing but TPKs is so morale-crushing as bleeding your way through a dungeon without finding any treasure, though booby prizes like trade goods and copper pieces are arguably worse. Monster treasure isn't (as far as I can tell) strongly linked to the rest of the economic system anyway. More thoughts on this later.
Another option which we did not consider, and which I am not sure the expected values on the treasure tables take into account rigorously, is the selling of magic items. We have been drowning in swords +1 in previous ACKS games, each of which is in theory salable for 5kGP and 5kXP. That adds up.
Finally, mid- or high-level dungeons could present gametime-denser sources of treasure than dragon lairs and treasuremap caches. Building dungeons is a heavy DM-side time investment, though, and suffers from a boom-and-bust cycle, where the players extract most of the easily-found treasure in a couple of high-yield early expeditions, then get frustrated in following low-yield expeditions. This requires either really aggressive restocking of monster lairs or continual generation of new dungeons. By comparison, hunting dragons or following maps seems like it should yield more consistent treasure; the only reason to fail to get the treasure on such an expedition is defeat or failure, rather than lack of intel. I also think high-level dungeons generally make less sense even than low-level dungeons, which serve as shelters for small groups of weak monsters. I suppose a sensible reason for a high-level dungeon to exist and not be dominated by a single entity or group would be the existence of a powerful magic resource (Midnight-style power nexes) or a gate to hell or some such thing which attracts or generates powerful critters, generating an unstable-enough situation that it doesn't just turn into a big lair for one most-powerful group of monsters.
In conclusion: getting to 9th level might provide you with the resources to build a flourishing domain out of nothing, but it also takes a long damn time and is not as amenable to fast-forwarding as the reinvestment approach.