When I was a Christian, leaders and missionaries were always talking about how Judaism uses the number three to represent God: the waving over the Shabbat candles three times, waiting for three stars in the sky on Shabbat, using a three wicked candle for Havdala, etc. They say it is because God is a trinity and that Judaism has just not "caught on" to the symbols God has given them to reveal his trinitarian nature. I never accepted this explanation from Christians, but then I also have no idea or explanation for why the number three is prevalent in the Jewish ceremonies. What are the reasons that "counting to three" is used so often in Jewish rituals?
Many numbers are used symbolically in Judaism - check out the our series of 338 (and growing) number-meaning questions.
The only number we use in reference to Gcd is ONE! The most important Jewish declaration is the Shma: Hear O Israel... Gcd is ONE.
Your missionaries' 3 examples of 3 are rather pathetic, to be candid:
the waving over the Shabbat candles three times,
There's no such a thing in Jewish law. Possibly some ladies have this habit, but it's not documented in mainstream Jewish law. (They are supposed to cover their eyes with their hands, so as not to see the candles until after the make the Bracha on them, for technical reasons.)
waiting for three stars in the sky on Shabbat,
This is a foolproof way for a layman to ensure it's actually night. And it's not 3 stars, it's 3 medium sized stars - meaning there are a few small stars already visible.
using a three wicked candle for Havdala
There's no such a thing in Jewish law. Possibly some people have this custom, but it's not documented in mainstream Jewish law. For Havdala one uses at least 2 wicks to create a torch, as opposed to a single-flame-candle. Many Havdala candles have 5, 6, 7 or even 8 wicks. I've never seen a 3-wicked one.
There are plenty of real examples where things are done or said 3 times - as doing or saying something 3 times is considered to make it definite.
Three is inherent in the human condition. Picture the TV character who has a tough decision to make -- a little angel version of himself appears on one shoulder, a little devil on the other. Altogether, three copies of the same face.
This is because whenever we feel torn in decision, we "hear" three voices -- A, B and the decider. Freud identified A and B as the Id and Superego, with the Ego in the driver's seat. Eric Berne (Transactional Analysis) had Parent-Adult-Child.
In Judaism, the Sefer haYetzirah identifies three aspect / categories of function of the soul. And from there, and Bereishis Rabba (5th cent CE) it reached both rationalists (such as Rav Saadia Gaon) and the Qabbalists. More about that at https://judaism.stackexchange.com/a/63814/1570 , so I see no need to repeat here.
(I would suggest that Freud, lacking belief in heaven and souls, could not believe in a neshamah having cravings for the spiritual as inherent as the body's -- really, the nefesh's -- cravings for sex, food, and other physical comforts. He therefore conceived of these drives inhering in a need to conform to one's social context, the Superego.)
Rav Saadia Gaon (Emunos veDei'os 2:4) says that our perceptions of G-d as "Living", Omniscient and Omnipotent led to the Christian trinity. A division in the human condition yields our perceiving G-d in multiple ways, and has nothing to do with G-d Himself. Again, the three-ness describes the human condition.
If any number is considered 'special' or 'holy' it would be seven, but even that is not consistent. The number three has a quality which is exemplified in Koheles 4:12: "A threefold cord is not quickly broken." This is the concept of Chazaka, that when something is repeated three times it is not a coincidence. (As in, 'I told you, not once, not twice, but three times!') Some prayers are repeated three times for this effect.
One thing must be clear, though. In Judaism there is never such a thing as having something represent Hashem. We see in Shmos 20:18-19 that Hashem says that now that we've seen how He spoke we should never make anything to represent Him. There is never a symbolism of an object representing Hashem, not as three and not as one. He created us to represent Him and that is our life's work.
The number three was always significant as many other numbers have always been long before christianity came along with their ideas. some examples are... chumash (5 books of moses, writings, and prophets (the sections of Jewish scriptures); the forefathers being Abraham, Isaac, and Jacob; the three parts of the intellect which are wisdom, understanding, and knowledge; a chazaka, and many others.
None of these examples have anything to do with counting to three but each have there own meaning for themselves and also happen to be grouped into threes.
the havdallah candle actually only needs two lights minimal and can even be accomplished by holding two matches together.
none of these things have anything to do with the trinity and missionaries pointing out their thing has the number three in it and so do many Jewish things is a form of manipulation. You could also find the number three relevant in other religions even besides christianity but that doesn't then mean the other religions have something to offer us just b/c they happen to use numbers as well. google the number three in Hinduism and you'll find a lot of examples.