As a form of literary rhetoric, the repetition of a word without any intervening words is called epizeuxis.
In rhetoric, an epizeuxis is the repetition of a word or phrase in immediate succession, typically within the same sentence, for vehemence or emphasis. A closely related rhetorical device is diacope, which involves word repetition that is broken up by a single intervening word, or a small number of intervening words.
As a rhetorical device, epizeuxis is utilized to create an emotional appeal, thereby inspiring and motivating the audience. However, epizeuxis can also be used for comic effect.
It includes several examples of the use of epizeuxis, including the following:
"Never give in — never, never, never, never, in nothing great or small, large or petty, never give in except to convictions of honour and good sense. Never yield to force; never yield to the apparently overwhelming might of the enemy."
“Isn’t it extraordinary that the Prime Minister of our country can’t even urge his Party to support his own position?! Yeah. Weak! Weak! Weak!”
The number of repetitions is not specified. So, there is no word that specifically means to repeat a word three times.
In another article that lists other forms of repetition, Wikipedia says this about repetition in general:
Repetition is the simple repeating of a word, within a short space of words (including in a poem), with no particular placement of the words to secure emphasis. It is a multilinguistic written or spoken device, frequently used in English and several other languages, such as Indian and Chinese, and so rarely termed a figure of speech.
Its forms, many of which are listed below, have varying resonances to listing (forms of enumeration, such as "Firstly, Secondly, Thirdly and lastly..."), as a matter of trite logic often similar in effect.