The number 666 has significance as the numerical value of the Hebrew verse: “Ata yigdal na koach Ado-nai — Now, I pray, let the Power of my Lord be great.” (Numbers 14:17). This was Moshe’s prayer invoking Divine Mercy on behalf of the Jewish People.
“Mosad Hayesod” cites the Vilna Gaon’s commentary on the Zohar that “the number 666 contains hidden within it exalted and lofty messianic potential.” No other explanation is offered there.
We do know that the number six represents the physical world. The Torah describes the creation of the universe as a six part, six day, process. Our ancient sources describe the universe as emanating in six directions — north, south, east, west, up, down — from a central point. All physical space and all physical objects have these six dimensions.
666 is six repeated three times. Repeating a concept three times represents the affirmation and strength of that concept. The number 666 could thus represent the strength and perfection of the physical world, which Judaism teaches will occur in the messianic era, when the physical world will reach its ultimate purpose, to be a vehicle through which the created experience the Creator.