What is the difference between 刚才 and 刚刚?
For example, 我刚才起床 and 我刚刚起床? Don't the both accomplish the same thing? 刚刚 means now and 刚才 means just now, right? Or does 刚刚 just less formal than 刚才?
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The difference between 刚, 刚才, and 刚刚 is described in references listed at the bottom of this answer. From these references (and using their examples), we find:
刚才 is a time-indicating noun (like 今天), whereas 刚 and 刚刚 are adverbs indicating "just recently" or "just" (perhaps not in relation to time). For example:
（✗）他把刚的事儿忘了。 [it should be 刚才 and not 刚]
（✗）这个教室刚才能坐下二十个人，多一个也坐不下了。 [it should be 刚 or 刚刚 and not 刚才]
刚刚 is basically the same as 刚, but is more immediate, and the emphasis is stronger. However, there's a set phrase 刚一 (and not 刚刚一):
（✗）她刚刚一回家，就开始打电话。 [it should be 刚一 and not 刚刚一]
Unlike 刚才, we cannot use 刚 nor 刚刚 before the subject.
（✗）刚他还在这儿，怎么一转眼就不见了。 [it should be 刚才 and not 刚]
Unlike 刚才, we cannot use 刚 and 刚刚 with a negative:
（✗）你刚不在的时候，玛丽给你来电话了。 [it should be 刚才 and not 刚]
This is explained at the Chinese Grammar Wiki: you can say something didn't happen just now (刚才), but you can't say that something just didn't happen (刚).
The time-indicating noun 刚才 cannot be preceded by another time-indicating noun, nor followed by a word of duration.
There are a few other differences in these references. In particular, see also the Chinese Grammar Wiki's discussion of the use together with 了.
The precise phrase 我刚才起床 arose on Lang-8, and it was corrected in a number of ways:
我刚才起床。 [i.e., it was left unchanged]
When leaving it unchanged, the author wrote:
Basically I do not think this sentence has to be corrected, but the two corrected sentences sounds more local.
So perhaps it's not strictly grammatically wrong, but sounds incomplete. It's possible to say comparable things like 我8点起床.
Both 刚才 and 刚刚 mean "just a moment ago"
The difference is: "刚才" is a two characters compound word, made up of 刚(just; moment ago) and 才(just), meaning "just a moment ago"; and 刚刚 is the reduplication of the adverb 刚(just; moment ago).
Reduplication an adverb put emphasis on it. For example, "我刚起床" alone already express all the meaning needed to be expressed, reduplicate "刚" and write "我刚刚起床" put emphasis on the adverb, make it stronger.
Since 刚刚 is a frequently used term, it is treated more like a compound word
Also: both "我才起床" and "我刚起床" has the same meaning of "I've just got up from bed"