By way of introduction, Sparta was actually kind of lame. Brett Devereaux of Unmitigated Pedantry does a good job covering this in this series, but to summarize: they cruelly indoctrinated children, kept 93% of their population in brutal slavery, and disparaged all other benefits of civilization (arts, literature, etc.). None of this even translated into military success; they had a victory rate of lower than 50%. Our Sparta-fetish comes from propaganda they carefully maintained.
My question is whether these kinds of extreme societal investments in military ever did pay off. Rome was obviously extremely effective in war and had a pretty martial culture, but they also had room for lots of other endeavors. I'm thinking about societies where military was the absolute center of everything (obligatory Klingon reference).
When those societies existed, did they ever actually perform better militarily?
While it doesn't hit the main question here, there's an excellent set of comments by /u/Iphikrates that should clarify the Spartans' reputation, as it wasn't so simple as "they were the best" or "it was all propaganda" https://www.reddit.com/r/AskHistorians/comments/6rvusy/comment/dl8ns8q/
In addition to the thread already linked by u/kdfsjljklgjfg, I would also like to highlight my older answer here, which explains why some modern scholars (including the world's most prominent authority on Sparta) believe that Sparta wasn't really all that militarised to begin with. When we look at Sparta, we're not looking at a society that minmaxed on militarism but didn't get the expected result. We're actually looking at a fairly ordinary Greek society that was the subject of much myth-making by outsiders because of a few notable achievements (mainly internal political stability). I cannot speak for all of history, but my baseline assumption is that something similar will apply to other societies that have traditionally been pigeonholed as "military cultures". Did such cultures ever really exist?
About states that did maximize for military, Qin, and to a lesser extent the other states of the Warring States period were highly militarized, and the extant texts of that period really do read like they are minmaxing military power. Qin would be quite successful in winning wars! It looks very different from even popular perception of Sparta, though.
I've touched on it a bit here and here, I'll rewrite it a bit below to be more specific to this question.
The period following the collapse of the Zhou hegemony saw widespread warfare, and when combined with new technologies a drastic shift in the organization of states. The development of crossbows, cast iron tools leading to higher outputs in farming enabled new large scale infantry armies which would eclipse the previous aristocracy centered chariot warfare. Centuries of bloody conflict led thinkers to try to imagine ways out, and led to the slow development of legalist thought which would dominate the bureaucracies of many of the states. Recent excavations at Shuihudi and Liye give us some insight into how these states worked in practice, and simultaneously refute later Han condemnations of insanely brutal, arbitrary punishments, but also that these reforms were real, and reached throughout society. The name legalism itself came later in the Han dynasty, as a useful shorthand for the reformers and thinkers who would design the new, powerful, centralized states of the Warring States. Rather than the more utopian thought of Mohism or Confucianism, they were focused on practical measures to create a "a rich state and a strong army", one strong enough to unify the world and end conflict.
These reforms took place in many states over a long period of time, but were taken to the farthest degree in Qin, starting before him but most famously embodied in the reforms of Lord Shang. These reformers were explicitly trying to create as powerful of a state as possible. They took a fundamentally negative view of human nature: that humans are lazy and greedy, but instead of trying to reform this nature as say, the Confucian Xunzi would, sought to create a state that would create incentives through rewards and brutal punishments that would lead to a powerful state. As people didn't want to fight, they should be incentivized to through closing out other avenues, rewarding soldiers, and executing deserters, as people didn't want to farm, other avenues such as merchants or scholars should be mercilessly squeezed.
As it is said in the Book of Lord Shang (translation by Pines):
Farming is what the people consider bitter; war is what the people consider dangerous. Yet they brave what they consider bitter and perform what they consider dangerous be- cause of calculation [of name and benefit]. Thus, in [ordinary] life, the people calculate benefits; [facing] death, they think of name (= repute). One cannot but investigate
whence name and benefit come. When benefits come from land, the people fully utilize their strength; when name comes from war, the people are ready to die
This is explicit in how they designed the system of meritocratic ranks. Rank would be available to anyone, including peasants, based off of success in battle, with the presentation of enemy heads or ears leading to promotion. In order to properly motivate people, ranks could be passed on to children, but with a reduction in rank, quite drastic for much of the high nobility, but the exact same rank could be passed if the owner died in battle.
The opposite was also true. Failure in battle would be monitored through pervasive social surveillance: family and fellow soldiers would be punished as well, and rewards given to people who informed on others. Such surveillance was extended throughout society to ensure that people dutifully followed their tasks, with the population divided into mutually responsible blocks of families that would be expected to spy on each other and inform of any misbehavior. Punishment was varied, but often meant forced labor, in anything from building projects to industry.
This apparently led to a ferocious army, as commented on by people from other states, with Qin soldiers fighting over enemy heads in the aftermath of victorious battles. It apparently didn't, however, lead to the type of glorified militarism seen in say, movies about Sparta: Legalist reformers thought that people would naturally be disinclined to fight, and would therefore need to be coerced into it. Evidence from excavations suggests that lower ranks were quite widespread by unification, with a quarter of the population holding ranks giving them status as low nobility. They would be granted immunity or reduced duties from corvee or conscription, lower tax rates, and as they ascended laborers of their own. There is a recorded, but probably very rare case of a slave ascending to a general. At the same time, forced labor as punishment also pervasive from agriculture to industry.
The reforms were far more reaching than just the ranks system, and extended into agriculture, industry, and how the population was mobilized. In order to maximize agricultural output and weaken aristocrats, widespread land reform was enacted, creating freeholding households and granting land allocations directly to the peasantry, which would be responsible for both conscription as well as taxation by the state, instead of a aristocracy between the central government and the peasantry. To maximize state control of peasant households, the peasantry was divided into nuclear households and prohibited to have adult males coinhabiting, though in practice a son (generally the eldest) was allowed to live with and support his parents as they got old.
To further increase agricultural output, large scale irrigation projects were undertaken, rewards and rank given out in some cases for high agricultural productivity, and the state pushed widespread use of iron farming implements and cultivation of wasteland. This was taken to an extreme in Qin's colonization of Sichuan, where the local elites crushed, and lacking any local aristocracy to push back, a massive irrigation project, that shapes Sichuan's geography carried out. Settlers were encouraged to move through rewards, convicts turned into settlers and forcibly relocated, and proto industrialists incentivized to open mines and foundries and so on.
Aristocrats still existed, but were brought more firmly under state control. They were granted labor by the state according to their rank for both agriculture and household purposes, but the state maintained their primacy to this, and would have the ability to arbitrarily pull labor if they had greater demands or punish those who mistreated convicts too badly. In one case, a county had a death rate of something like 20% in forced laborers, and pretty much every important official got hauled off to trial.
Finally, the state exercised great control over industry and merchants. Though the Book of Lord Shang is very negative towards merchants and artisans, many of the other surviving legalist works are less negative, and simply seek to ensure that the output of them is structured towards the goals of a strong state. Qin seems to have worked that way in practice. There were both large, centralized state workshops with expert artisans and large scale use of forced labor, as well as a distributed network of private factories run by industrialists under state supervision. The later can be seen in the manufacture of crossbows, where a private network of factories in a cellular manufacturing process was broadly overseen by government officials. The parts were probably not truly interchangable, but showed a much higher degree of standardization than random, and were all stamped to indicate where they came from, so defects could be traced. Mining was similar, though may have not used forced labor as it was hard to control in remote areas that mines were often located in, rather run by private enterprises under government supervision.
This all led to a highly militarized state designed to optimize for military power. Highly productive agriculture supported large, conscript armies, which were motivated through a brilliantly designed system of rewards and punishments, and supported by a network of state supervised resource extraction and large industry. This state, in the form of Qin, would succeed in ending the Warring States, but the extreme levels of militarization would fade after Qin's collapse and into the Han. Maximizing agricultural output, manufacturing a ton of crossbows and iron farming implements, then giving them to peasant conscripts who, through success on the battlefield can ascend to the nobility, all looks very different from, say, our image of elite Spartan warriors.
Sources that I don't think are in my above posts:
“Social Engineering in Early China: The Ideology of the Shangjun shu (Book of Lord Shang) Revisited", by Pines.
He also has a recent translation of the Book of Lord Shang, and I think wrote quite a bit of the very good overview of legalism in the Stanford Encyclopedia of Philosophy.
"Empire-Building and Market-Making at the Qin Frontier: Imperial Expansion and Economic Change, 221–207 BCE", Maxim Korolkov is a very good look at what this looked like in practice.
To sidestep the other discussions, could you perhaps define what you mean by "militarized"? If you just mean societies that put a high value on martial pursuits you could include many Central Asian steppe societies, and obviously the Mongols are perhaps the most succesful conquerors in history, but I am not sure I would consider that "militarized". The degree to which Rome was "militarized" is also somewhat complicated, it is true that during the high empire it kept a large, professional standing army which is possibly unique in the history of pre-modern empires, but the actual proportion of the population under arms was comparable to the modern United States (about 0.5%).
Beyond that, the historian Bret Devereaux wrote a series of blog posts related to this that you might find interesting.