Good question. Let's look at the definition of addition. Addition is defined as a binary operation. What this means is that it is a function ##+:\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}##. So to ##(a,b)##, we associate ##+(a,b)##. Since this looks weird, we just write it as ##a+b##.
From this definition, we see that something like ##a+b+c## does not make sense. We can't define it as ##+(a,b,c)##, since ##+## only operates on ##2## terms. However, the following is defined:
(a+b)+c
In more primitive notation, we have ##+(+(a,b),c)##. So we let ##+## operate on ##+(a,b)## and ##c##. But we can also define
a+(b+c)
which of course means ##+(a,+(b,c))##. And it turns out that both are equal. So, we have
+(+(a,b),c) = +(a,+(b,c))
It is for this reason, that we can invent the notation ##a+b+c## to mean the value of the above. This is an abuse of notation. Purely rigorously, we can't write it like ##a+b+c##. But of course, the abuse is very convenient.
There is an operation, called the Lie bracket (denoted by ##[a,b]##). What it is, is of no importance. But it is also a binary operation. However, it is not associative in general. Thus we have
[a,[b,c]] \neq [a,[b,c]]
This is a major example of a nonassociative operation. Because it is nonassociative, we do not write it as ##[a,b,c]##. We always write it as a binary operation.
The cartesian product of sets is a bit more subtle. There, it does not holds that
A\times (B\times C) = (A\times B)\times C
So the operation fails to be associative. However, we do write it as ##A\times B\times C##. The reason is that there is a natural bijection between the two sets. That is, we can identity the element ##(a,(b,c))\in A\times (B\times C)## with the element ##((a,b),c)\in (A\times B)\times C##. This is a canonical or natural identificiation. So when we write ##A\times (B\times C) = (A\times B)\times C##, we always keep this identification in mind.