What is the meaning of 'compounded' & 'uncompounded'?

While @ruben2020 answers this question well, Ruben does not explore the meaning of 'uncompounded' with as much rigour as Ruben explores the category of 'compounded'.

Ruben mentions 'uncompounded' only within the scope of Nirvana, and indeed states that Nirvana is the only phenomena which is not compounded

Nibbana is the only phenomena which is not conditioned and not compounded.

Ruben's argument is based upon a citation from Ud8.3 which, while (correctly) stating that Nirvana is uncompounded, does not claim that Nirvana is exclusively so.

If we consider that compounded phenomena are products (and products are necessarily impermanent and dependently arisen) then we can also identify many more things than nirvana which are uncompounded.

Non-existent things

Non-existent things are not compounded, because they do not exist. For example, an inherently existent self, or a fantastical hobbit. (Q: How is a hobbit uncompounded? A: While the hobbit (in middle-earth) is fictionally compounded, the consequences of the actions of a hobbit are bound only by the imagination of the author. Therefore, while the hobbit is depicted as a compounded phenomenon, a hobbit is a mere fiction. this does not detract from the impact that the fiction has - the fiction (as a narrative) is an existent, compounded phenomenon, but the contents of that narrative are non-existent, and therefore uncompounded).

Non-existent things do not intrinsically exist - because they do not exist.

Permanent phenomena

("Permanent" here does not mean 'everlasting' - it means not subject to momentary change which leads to it's disintegration.)

Permanent phenomena are uncompounded because they are permanent (not subject to momentary change), while compounded phenomena are impermanent. Permanent phenomena are existents. Their designations might be compounded, but the referents are uncompounded.

There are many ways of categorising permanent phenomena, but in the end all of them are absences. Here are a few (and this is not an exclusive list):

Cessations (nirodha) are permanent phenomena. Nirvana is one such cessation, being the cessation of suffering and its causes. Just because a cessation has arisen, does not necessarily mean that we do not feel the consequences of such a cessation. We continue to receive the benefits of Lord Buddha, despite his winning the cessations (e.g. the cessation of ignorance) associated with Buddhahood.

There are many other cessations: Anything that only exists in the past. We continue to be affected by them, but they have ceased. So cessations are not limited to spiritual achievements.

Temporary permanent phenomena This is the absence of something that might return. For example, a candle may be blown out. The subsequent absence of flame is a permanent, but there is no cessation, as the conditions are still available for the candle to be lit once more.

Mere Absences The absence of an elephant in this room is a permanent: If it (the absence) were compounded, it (and an endless number of other absent things) would decay until at some point the absence would cease. If absences were impermanent, we could wait until the absence of 1 million ... No! 100 billion dollars in our pocket had disintegrated!

***Suchness / Tathātā *** This is an important term - used in, for instance, "Tathāgata" The naturally absence of inherent existence is a permanent. The absence of inherent existence does not require a Buddha (or others) - it is the final truth, but it is an absence - so is not composite. (Theravada consider tathātā to be conditioned, but I do not know how)

Space The space between particles - the complete absence of matter - is a permanent, being non-composite and non-disintegrating. (It might be that there is no actual absence of matter in space - but this doesn't affect our path to enlightenment either way).

Permanent phenomena depend upon designation and do not intrinsically exist. This is true of cessations such as nirvana.

Everything else (ie, other than non-existents and permanents) is a compounded thing, subject to dependent-arising, and does not intrinsically exist being a composite, a dependent-arising and dependent upon designation.

An important distinction between non-existence and absence is that absence does not necessarily indicate the possibility of the entity that is absent, whereas non-existent entities cannot exist. An example is inherent-existence, which is naturally absent because it is a fiction. Being a fiction entails that it cannot exist. However, the absence of a flame does not indicate the same level of negation. (I am not 100% sure that all non-existents can be established as 'impossible to exist' - but there are plenty of examples where that is the case.)

sankhara means a construct, a conditioned thing, a relative thing, or a compound, meaning something made of parts and existing only dependent on other phenomena.

conversely, Nibbana is asankhata, meaning not constructed, unconditioned, absolute, so not a compound, not made of parts and existing independent of phenomena.

The terms "conditioned" and "compounded" are often used to explain the term "sankhara".

Conditioned means phenomena which are affected or caused by each other. For e.g. a tree is not just a standalone object. It is dependent on sunlight, water, air etc. Space and time too are conditioned as we can see in Einstein's Theory of General Relativity.

The following writing entitled "The Fullness of Emptiness" by Thich Nhat Hanh explains this nicely:

If you are a poet, you will see clearly that there is a cloud floating in this sheet of paper. Without a cloud, there will be no rain; without rain, the trees cannot grow; and without trees, we cannot make paper. The cloud is essential for the paper to exist. If the cloud is not here, the sheet of paper cannot be here either. ....

If we look into this sheet of paper even more deeply, we can see the sunshine in it. If the sunshine is not there, the forest cannot grow. In fact, nothing can grow. Even we cannot grow without sunshine. So we know that the sunshine is also in this sheet of paper. .... And if we continue to look, we can see the logger who cut the tree and brought it to the mill to be transformed into paper. And we see the wheat. We know that the logger cannot exist without his daily bread, and therefore the wheat that became his bread is also in this sheet of paper. And the logger’s father and mother are in it too. When we look in this way, we see that without all of these things, this sheet of paper cannot exist.
The Fullness of Emptiness

The term compounded explains how phenomena is composed of other phenomena and are not standalone. For e.g. a tree has water and gases in it. Molecules can be broken down into atoms and atoms can be broken down into subatomic particles. And all of these are also conditioned.

The sutta below provides an explanation of what is meant by compounded but also explains what is convention.

A chariot is never standalone. It is an assemblage or composition of different parts, like wheels, axle etc. The term "chariot" is therefore a convention that we use to describe this composition. If one wheel is missing, or replaced, it is still called a "chariot". But if all parts are disassembled, then we don't call it a "chariot" anymore.

So, the term "chariot" is a convention used to describe an object that is compounded by many parts.

Similarly, a "being" (satta) is the convention used when the five aggregates of form, feeling, perception, consciousness and mental formations are present together and functioning. A "being" is therefore compounded.

Then the bhikkhuni Vajira, having understood, "This is Mara the Evil One," replied to him in verses:

Why now do you assume 'a being'?
Mara, have you grasped a view?
This is a heap of sheer constructions:
Here no being is found.

Just as, with an assemblage of parts,
The word 'chariot' is used,
So, when the aggregates are present,
There's the convention 'a being.'

It's only suffering that comes to be,
Suffering that stands and falls away.
Nothing but suffering comes to be,
Nothing but suffering ceases.

Then Mara the Evil One, realizing, "The bhikkhuni Vajira knows me," sad and disappointed, disappeared right there.
SN 5.10

The sutta below gives the analogy of a stringed musical instrument, called the lute (vina). It can produce music. But if you break down the lute into its constituent parts and air, you cannot find music anywhere. Music is both conditioned and compounded by the different parts of the lute, and even the actions of the lute player. Music is made up of rythmic sound composed of movement of air. And this movement is dependent on the movement and vibration of parts of the lute.

Similarly, the mental idea of the self is conditioned and compounded by the five aggregates of form, feeling, perception, consciousness and mental formations. Dependent origination (paticcasamuppada) is the Buddha's explanation of how the birth of the mental idea of the self is conditioned by the five aggregates. But it's also compounded, because the mental idea of the self is also a type of thought, a mental formation.

"Suppose there were a king or king's minister who had never heard the sound of a lute before. He might hear the sound of a lute and say, 'What, my good men, is that sound — so delightful, so tantalizing, so intoxicating, so ravishing, so enthralling?' They would say, 'That, sire, is called a lute, whose sound is so delightful, so tantalizing, so intoxicating, so ravishing, so enthralling.' Then he would say, 'Go & fetch me that lute.' They would fetch the lute and say, 'Here, sire, is the lute whose sound is so delightful, so tantalizing, so intoxicating, so ravishing, so enthralling.' He would say, 'Enough of your lute. Fetch me just the sound.' Then they would say, 'This lute, sire, is made of numerous components, a great many components. It's through the activity of numerous components that it sounds: that is, in dependence on the body, the skin, the neck, the frame, the strings, the bridge, and the appropriate human effort. Thus it is that this lute — made of numerous components, a great many components — sounds through the activity of numerous components.'

"Then the king would split the lute into ten pieces, a hundred pieces. Having split the lute into ten pieces, a hundred pieces, he would shave it to splinters. Having shaved it to splinters, he would burn it in a fire. Having burned it in a fire, he would reduce it to ashes. Having reduced it to ashes, he would winnow it before a high wind or let it be washed away by a swift-flowing stream. He would then say, 'A sorry thing, this lute — whatever a lute may be — by which people have been so thoroughly tricked & deceived.'

"In the same way, a monk investigates form, however far form may go. He investigates feeling... perception... fabrications... consciousness, however far consciousness may go. As he is investigating form... feeling... perception... fabrications... consciousness, however far consciousness may go, any thoughts of 'me' or 'mine' or 'I am' do not occur to him."
SN 35.205

Nibbana is the only phenomena which is not conditioned. It is also not compounded. This can also be inferred from the three marks of existence, as found in this answer.

“There is, mendicants, an unborn, unproduced, unmade, and unconditioned (asaṅkhataṁ). If there were no unborn, unproduced, unmade, and unconditioned, then you would find no escape here from the born, produced, made, and conditioned. But since there is an unborn, unproduced, unmade, and unconditioned, an escape is found from the born, produced, made, and conditioned.”
Ud 8.3

The term asaṅkhata seems to mean unconditioned and also uncompounded, according to the PTS Pali-English dictionary:

Sankhata Sankhata [pp. of sankharoti; Sk. saŋskṛta] 1. put together, compound; conditioned, produced by a combination of causes, "created," brought about as effect of actions in former births S ii.26; iii.56; Vin ii.284; It 37, 88; J ii.38; Nett 14; Dhs 1085; DhsA 47. As nt. that which is produced from a cause, i. e. the sankhāras S i.112; A i.83, 152; Nett 22. asankhata not put together, not proceeding from a cause Dhs 983 (so read for sankhata), 1086; Ep. of nibbāna "the Unconditioned" (& therefore unproductive of further life) A i.152; S iv.359 sq.; Kvu 317 sq.; Pv iii.710 (=laddhanāma amataŋ PvA 207); Miln 270; Dhs 583 (see trsln ibid.),