The society in question would be composed of a single non-human species that evolved naturally. They are lead by a king, alongside multiple dukes who manage smaller portions of the kingdom. To distinguish these leaders from the rest of the populace, they are far larger, with the dukes being twice as big, and the kings being three times larger. They also have other features, such as more pronounced natural ornaments and weapons. However, there is one wrinkle in this system: The kings and dukes are elected by the populace. What features could this species have, that could be co-opted by the society to make the kings and dukes so impressive?
It’s a matter of diet.
A certain food (fruit, vegetable, bodily excretion) is reserved for royalty. This food triggers or maintains the extra growth as it is calorie rich and contains vital nutrients/hormones to trigger the process. It could just be fat and muscle mass that’s gained (like with gorillas, whose alpha gets the best food and is therefore the strongest), or you could opt for more significant structural changes. In the former case a lack of this foodstuff could lead to once-elected kings losing their size after being deposed. In the latter case a king is always identifiable by their extra features. In either case additional features (skin colours, hair, extra cosmetic flaps of flesh) could develop.
When your species ‘vote’ they do so by finding (or possibly exuding) this foodstuff and giving it to their preferred candidate. The dukes then give a tithe of their hoard of royal food to the king (add as many layers as you like). The candidate with the most food will be stronger and thus be able to seize production of the total food from their rivals once all the ‘votes’ are cast (in more civilised times this can be replaced with actual voting).
It’s easy to see how this behaviour could evolve in a caste driven eusocial species (have a look at the Naked Mole Rat for an example: Their queens get much bigger and fatter once they are selected, and are selected late in life), and once sociology takes over it’s easy to see how it could be maintained even if it’s detrimental to the proletariat. Kings/dukes could even dole out small portions to trusted bodyguards/enforcers to ensure the proper distribution of the foodstuff both during and outside voting season.
After all: who wants serfs to be as strong as their lords?
This reminds me of certain fish, where if the dominant male dies, the dominant female transforms from female to male.
Everyone in your society is born female. Your royalty could be the males, the large number of females consisting of most of the populous giving a feedback signal to these males. The males would need to keep interacting with the people and staying in touch to get the feedback signals and remaining male.
The election system wouldn't even need to be open - the more females that pay homage to a male, the more differentiated he becomes, and the more pronounced the sexual differentiation gets. Or, if you want an election, then the females place their approval scent on stones and drop them in a bucket for a given male. On election day, the chosen individual is permitted to open their bucket and inhale the differentiation signal, while the loser has their bucket rinsed out. You could even have males converting back into females if their disapproval ratings get too high.
It is because in their society, the taller somebody is, the more votes he/she gets
It is true, nominally these aliens are democratic. But they have the strange pattern that people trust only tall people. Literally, as among humans politicians compete trying to look more honest or more capable than their opponents (or just shouting louder at political talk shows), in the alien society people try to look taller than their opponents.
This means that tall kids are soon selected by political parties and grown to become kings or dukes (so they are given better food and maybe some growth ormones). Add to the mix that there are some very rare genetic variations that allow some alien to grow a lot more than the average.
At every election, the taller partecipant is invariably the one who is elected for a position. The loser than has to concede, and run for a lesser position. In the end the society will settle to a situation where the taller somebody is, the higher political office he/she will chair.
Some time ago I read that in certain mice when two males fight for their role in the hierarchy, the winner and the loser immediately change their body odor after the fight to reflect their new status.
You can have a similar system, based on hormones, to reflect the status of the winner. The winner will get a boost of hormones and as such
they are far larger, with the dukes being twice as big, and the kings being three times larger.
Hormones and gene expression
Genes dictate a lot for enzymes and hormones. These in turn help with growing the body. These determine when to grow and what. Look at the development of babies through adulthood. Several growth periods where specific things are growing. Babies their brains create many connections and prune them, apoptosis in the womb removes the webbing between fingers, arms and legs frow first a lit during puberty, after which the rest follows, if not to speak of getting to sexual maturity. These things on their own can be influenced, although for humans it's quite limited. You can stop a lot of processes with malnutrition for example, which is why some girls stop their menstruation cycle when malnourished for example.
To have another example, metamorphosis of butterflies can fully transform a being into something else.
Interestingly there is also gene expression. If you have a well fed life, other genes will activate than if you're malnourished. If you get depressed, you make less of certain hormones and thus stay depressed. Both are in part gene expression. Certain genes activate because some situation is happening. These triggers can happen quite suddenly, as a single psychosis or trauma can already result in some gene expression.
That changes happen is only part of what you want. You want a trigger. There are actually a lot of creatures that can change a lot thanks to environmental triggers. Frogs that can turn itself from male to female or vice verse. Ants smelling insufficient soldier ants and morphing into the stronger version. Certain reptiles(?) that stay in a fish form and can reproduce, but when the pool dries up they start growing legs and can move onto the land. Jelly fish can recert to earlier stages of their life when there isn't enough food available. Some bees can be fed the royal jelly and transform into queens. There's triggers enough for a wild variety of results.
The transformation to larger versions seems the easy bit. The trigger is more tricky. Having been chosen at an election can give a huge boost in hormones. These can immediately start hormonal loops, which can in turn drive gene expression. The gene expression are the building plans for the next phase, much like puberty or a metamorphosis. These can be continued until the correct size is achieved.
The mechanics behind the evolution of elected transformation can be in the psychology as well. In earlier days it might've been the strongest or best to lead that a group of these creatures instinctively chose. This started to take more ritualistic forms until in the modern times the hormonal boost could be gotten from elections.
In a nutshell
The creatures can have evolved with several sizes, which is not uncommon in nature. Hormones and gene expression can determine when a creature needs to transform and what it should transform into. The trigger can simply be the election, which thanks to generations of evolution results in a massive spike of some hormones, which trigger positive loops and gene expression to start the transformation.
The society has a psychic field that makes leaders larger.
The more followers a person has, the more the psychic field augments them, increasing their size, weapon quality, and suitability to lead as it makes their body secrete growth hormones and other growth things. Elections make you more popular and so increase your resources.
It's fairly common in elections to have 'debates' where the two candidates duel for their honor. Whoever wins will impress the population and get larger.
A king has to win duels against multiple dukes to achieve their position and so they tend to get very large.
Your alien species is one of intelligent, modular, colonial, biodemocrat organisms!
The individuals of your alien species aren't discrete beings at all - instead, they are giant colonies of individual smart organisms that bundle together to "create" an working unit. Those units are highly cohesive and specialized but able to work alone in case of need, each bit having limited brainpower that it contributes to the bigger colony-being.
Those bits can also be willingly detached and traded with one another, as they are fully compatible between different individual beings. Exchanging parts with one another isn't only possible but part of their culture - swapping one of your notahearts with one of your loved one could be seem as a marriage ritual, while exchanging your Top Left notahand with your best friend is a signal of deep friendship.
Lending or giving out parts of oneself to others is also something that those beings can do. When not replaced after some time, the detached part eventually grows back, if the "giver" has enough nutrients in their diet.
The person that receives those parts without giving anything in exchange can add them to their body for improvements - extra arms, added muscle-like fibers, an extra light sensor on your shoulder... you name it. Those beings are modular by nature, and they can get rather large and full of extra features if they have a large enough fanbase. However, their cores can only handle so much safely - excess extra parts can only exist on a body for so long before they wither and fall off, needing a replacement if this "enlarged" being would like to keep their enhanced physique.
For this culture, people that stay at home lend some of their notamuscles to those that are going out to hunt. A female can take out her notaboob and slap it on her male partner when it is his turn to breastfeed. Healers can take out some redundant parts and shove them into people that are hurt or missing bits because some accident, so the injured ones can survive until they can grow their own missing bits back. People can negotiate borrowing or buying high-performance or prettier parts from their peers for enhancing oneself, or even buy out artificially-engineered parts if their tech is advanced enough. Those aliens can literally pay someone to work out for them, swapping parts before hitting gym and them swapping back again once the training is over.
Back to your question, however. Dukes and Kings!
An elected "duke" or "king" receives tithe in the form of body parts from their subjects every now and then. They are hulking beings not because something inside them triggered this transformation, but because this lego-like society gave them the bits they need to grow. When it's time to give the seat to someone else, they stop receiving the tithe and eventually all the extra bits wither and fall off, returning the now ex-duke or ex-king back to their original form.
Thus, the dukes and the kings are literally made out of bits of their subjects, and just as strong as the people they care for are. If enough people are unhappy, they can stop paying tithe to the king or duke and instead provide for someone else, effectively booting out the previous governor and replacing it by sheer power of biodemocracy!
This is quite possibly the strangest concept for an alien species that I came up so far. I loved answering this question.
It's a byproduct of the geography, radiation or some similar external input that is present in the chosen capitals and not around the general populace. Possibly there's a place of worship in those places that emits the energy and the leaders frequent there more often than anyone else.
Be more orangutan
Dominant adult male orangutans go through substantial bodily changes, developing facial "flanges", throat pouches, and a more resonant chest cavity. They also become much more fertile. Other males tend not to develop these characteristics whilst a dominant male is in the area.
Interestingly, it seems that this is not triggered by anything hormonal, so (unlike queen bees developing) it is not down to body chemistry. Instead it seems to be triggered by calls from the dominant male. This suggests that the cause of the change is psychological, and a male orang will only trigger this change when it feels it has an opening. This makes evolutionary sense, because it signals to the dominant male that the non-dominant male is not a threat and hence a dominant male is less likely to attack in earnest. We don't know whether the change is psychosomatic and involuntary, or voluntary and can be triggered at will.
Your species may have evolved from ancestors with similar traits. Whilst their society has developed into a more formal social structure (whether with elected officials, feudal hierarchies or whatever), their physiology reflects their origins. The more dominant a male feels in the social structure, the more he will express these characteristics.
Your question now is whether your society should be entirely male-led. If you want equivalent dominant-female characteristics, they will not be the same as a dominant male. And you might want to be thinking about the societal implications of sexism like "How can we take her seriously as a leader when she doesn't have a throat pouch?"
An election involves individuals secreting a quantity of rularium and contributing it to that individual's preferred candidate(s). The rularium is consumed by the the candidates, and if they attain the appropriate bodily specifications within the allotted period after the election, they are the clear winners. Otherwise the election process continues until there is a clear winner.
"Anatomically different" doesn't necessarily mean extra limbs or different size. It could be something as simple as an eye, skin, or hair coloring that doesn't occur naturally, but could be triggered by a special dietary additive. This has the advantage of wearing off after they leave office and their diet returns to normal.
Presumably, such a society would have laws to preclude those not holding office from having access to the special formula. A plot twist might be that a non-elected character is impersonating an official via contraband access to the formula.
Upon winning an election, the victor quaffs a mysterious potion whose recipe is known to a small number of clerics. This potion contains a parasite that causes author's choice (antlers, colorful feathers, more prominent signals of sexual prowess, you name it) that signify health, wisdom and power. It sounds far-fetched, but it turns out ...