At some point, someone saw the kanji 協 and thought “nah, we need another 力 in there”. And so 協力 was born.
-”But sir, will it change the meaning?”
-”No. It will mean the same as the kanji by itself”.
Have you found other kanji that exaggerates?
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This is often because compounds usually allow for more variety and clarity than if everything was expressed with single kanji.
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Isn’t it also a case of the writing system coming (much) later than the spoken language? 
I don’t know about the origin of this one in particular, but it’s not outlandish to think that the spoken word existed well before the kanji - they needed writing symbols to match the meaning of the sounded きょうりょく and the phonetic of 協 would not suffice.
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I’ve also met quite a few examples of compound words made of kanji meaning the same thing. For example: 牢獄
Prison + Prison = Prison
Can’t help imagining the following dialogue:
– I’m in prison!
– You mean, like, prison prison or just something prison-ish?
– Prison prison. 牢獄。
– Man, this is bad then!
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Only yesterday WK taught me 刑務所… I guess prison-prison wasn’t enough? 
I hadn’t seen 牢獄 before - sounds like an older word? whereas this one might be a more “modern” term? 
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It’s necessary for avoiding confusion while speaking I think.
Instead of 牢獄 (rougoku, EN=prison), you could just use 牢 (rou, EN=prison) to mean the same thing. But if I just say ‘rou’, according to jisho, there’s a whole bunch of different kanji and meanings that use that reading, so it could lead to confusion. I could be talking about ‘prison’, or ‘offering ceremony held on the third day of the dog after the winter solstice’
But if I use 牢獄 (rougoku), there’s only one entry for that, so there’s no ambiguity over what I mean.
It might be the case for 協 (kyou, EN=cooperation). Kyou can have a lot of different meanings as well, but 協力 (kyouryoku) only has two, and only one of them is a noun.
I’m guessing this is one of the main reasons. But it doesn’t always apply, for example 身体 (shintai) means ‘body’, but ‘体’ (karada) on its own is also a pretty unique reading, so the disambiguation isn’t necessary. So maybe in that case it’s more to do with creating a different word for formal or literary /informal situations, or something like that?
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A nice one is ‘forest’.
You can have a forest with 3 trees 森 (もり) or with 5 trees 森林 (しんりん).
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Oh, level 8 just answered my question!
“Wonder about the difference between 体 and 身? 体 refers to physical human and animal bodies, often describing their condition, appearance, etc. While 身 can also mean “physical human body,” it can also refer to the self as an individual and be used to indicate one’s social standing or position. That’s why you might see 身 used in idiomatic phrases like 身につく, which refers to acquiring or mastering something, like a skill.”
so it seems like 身 is preferred here to clarify that it’s the general idea of the human body only (the systems and structure of the body, as opposed to the physical ‘object’ one sees)
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i feel like 協力 was probably originally some sort of way of flattering someone on how much help they were giving (compare “you’re a big help”), or doing so before they actually helped to try to get them to do it. so 力 was probably emphatic
there’s also the bit about the need for disambiguation which is probably why 2-character combinations became more common than simple 1-character ones - but i’m guessing they’re not exclusively for disambiguation and still have some semantic meaning to them
something like 牢獄 can make sense because 牢 originally seems to have meant something like impenetrable - an attribute for something that you probably couldn’t get through or into, such as a mountain (獄 seems to be more general than jail, and i think it refers to a place where a person is held via force somehow. so just regular 獄 maybe could’ve referred to being tied up - if you were the emperor and you wanted to send prisoners off to 獄, you might want to qualify that you would like the 獄 in question to actually be a solid, set structure, rather than just guards staring your prisoners down. if someone decided they wanted the prisoners back, they might bring over their army to kill your guards and get the prisoners back - if you have a 牢獄, it’s more defensible in the event of such a thing. these days it probably doesn’t matter, but people will tend to use words that are easily disambiguated to avoid confusion, so they’ll pick two-kanji rather than one-kanji words)
or at least that’s my analysis - no clue if it’s accurate or not
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I think for two kanji onyomi compound words it’s helpful to consider the history. These are mostly loanwords from Chinese that came along with the Chinese derived kanji readings that they use. So the question of “why are 協力 and 牢獄 words?” is as much down to “why are these old Chinese words that got loaned into Japanese multisyllabic and multi character?”, as to anything about Japanese. The mangling of the Chinese pronunciation probably also increased the number of homophones in onyomi, making single character onyomi words less likely to be used as loanwords. (Japanese syllable structure, especially at that time, was much less complex than Chinese of the time.) As loanwords, they’re also naturally likely to be used when the author wanted to be precise about something in a way they felt the native Japanese word couldn’t cover.
(The other main source of two character onyomi words is that there were a lot of coinages of new words in Meiji era Japan at the end of the 19th century, usually to translate words from Western languages and texts. Here the answer is that hundreds of years of familiarity with the Chinese-derived loanwords made this kind of new word (the multi character onyomi) easy to accept and at least vaguely indicative of intended meaning, in the same way that latinate compound words for scientific concepts could enter English.)
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