The fact that the three-part list is such a commonly used rhetorical technique often raises the question of what it is about the number 3 that’s so special. It’s sometimes suggested that there must be something magical or mystical about it. But there is, I think, a much more rational explanation.
Its most likely source is the commonest form of human communication, namely everyday conversation. In the research literature of conversation analysis, there is the idea of there being an implicit or tacit ‘economy rule’ that works along the lines of “say things as briefly as you can unless you have a good reason for doing otherwise”.
So when we complain that someone is ‘long-winded’, ‘likes the sound of his own voice’ or ‘is always hogging the conversation’, we are in effect noticing and complaining that he’s breaking the economy rule. And the fact that there are actually words in the English language, and probably in other languages too, for referring to such rule-breaking behaviour (e.g. ‘verbose’ and ‘garrulous’) means that it’s something that happens (and is complained about) quite often.
We also know something about how three-part lists work in everyday conversation, thanks to a fascinating paper by one of the pioneers of conversation analysis, the late Gail Jefferson. Among other things, her detailed empirical studies showed how frequently they occur in conversation and how they are often treated by others as indicating that the person who produced the three-part list has finished and that someone else can now start speaking without fear of being accused of interrupting.*
As for why so many lists come in threes, the most likely explanation is that 3 conforms to the economy rule, because the arrival of a third item is the first point at which a possible connection implied by the first two is confirmed, and has the effect of turning a ‘possible list’ into a ‘definite' or 'complete enough' list’ of similar things.
For example, if my first two words are ‘Rose’ and ‘Lily’, I could be starting a list of flowers' names or a list of women’s names – but you won’t know for sure which of these it is until I add a third one.
If my third word is ‘Joanna', it becomes a list women’s names; if it’s ‘foxglove’, it becomes a list of flowers' names (and, if it had been ‘Botham’, cricket fans will hear it as a list of cricketer’s names - Lily and Lillie, the famous Australian fast bowler, may be spelt differently, but they sound exactly the same).
The arrival of the third item in a list is therefore the first and earliest point at which a possible list can become an unequivocal and unambiguous list of similar items, and you don’t need a fourth, fifth or sixth word to establish this. So, if you’re following the economy rule, you wouldn’t (and shouldn’t) add any more after the third one.
This is not to say, of course, that we can never use longer lists in conversation without risking complaints. Sometimes we want to convey a sense of ‘muchness’, as when we want to emphasise that something was much better or much worse than normal. “He went on and on and on” may be a fairly common way of complaining that someone was being ‘long-winded’, but “He went on and on and on and on and on” is to make a more serious criticism by depicting the violation of the economy rule as having been far more extreme than usual.
*Gail Jefferson (1990) 'List-construction as a task and a resource'. In: George Psathas, eds. Interaction Competence. Washington, D.C.: University Press of America: 63-92,