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What is the significance of the numbers 3 times 7 in the story of Lavan other than that Yaakov worked for Lavan for (almost) 3 times 7 years? Do these numbers have significance elsewhere that can be applied to here?
See further: Why did Ya'akov work for Lavan for so long?
The answer from Al Berko doesn't give actual sources for his answer and his explanation is difficult, at best to follow. But looking at the original question points to the answer to your question.
The Ohr HaChaim to Bereshit 29:18 comments on Yaacov offering to work 7 years for Rochel that a Tzaddik always subjugates himself seven. (עצמו שבע) This relates linguistically to the number seven, the verb to satisfy, and to making an oath. He says it relates to the Sod concept from Mishlei 24:16, שבע יפול צדיק וקם. That through subjugating for seven the adversary is satisfied and overturned by HaShem.
This same idea is mentioned in Ba'al HaTurim to Bereshit 46:2. There, the Tur mentions that when writing the word לישראל in a sefer Torah, seven crowns are placed upon the letter Shin as a sign that G-d would save Yaacov from 7 troubles.
The letter Shin has 3 arms and also has a Mispar Katan value of 3.
It is unclear what Al Berko is referring to in regard to the number 3 except that it may be a reference to the concept of chazaka. That repeating something three times establishes it as a fixed condition.
He points to the idea that Yaacov was 84 years old when he actually married Leah and Rochel. When divided by 7 it results in 12, which could be an allusion to his comment that Yaacov stole the blessing of his brother, Eisav.
According to Berko in the name of the Ari z"l and the Ba'al Shem Tov, that included that Eisav would have been the father of the twelve tribes of Israel. Specific sources for those two references were not provided.
And if 84 is divided by 21 (3 times 7), the result is 4 which could be a reference to G-d's four letter name, who saves Yaacov from his seven troubles.
Thank you for daring asking the question in big. As I understand, this site focuses on the Peshat level, and I'm getting a lot of critics for mixing the Drush in.
My answers are based on (my understanding of) what is called Derech Ha'Avoda (in the way of the Divine Service(?)), which is one of "branches" of the Drush (Kabboleh of you wish)[1]. It implies, that this world in general and every single level of it, just as every single person (Neshomah), has a mission to fulfill. The Mitzvos (explicit and implicit) and the Torah study are the ways to fulfill the missions, either by the way of performing it (think sacrifices) or studying about it (נשלמה פרים שפתינו). A proper fulfillment of a mission gets us to the next level, just like in video games, and the mission itself becomes "obsolete".
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Usually those missions target a Tikkun (תיקון "restoration", maybe you use a better translation) of a certain aspect of the world after it was damaged by a sin. For example - the primordial sin (נחש) led to obligation of marital relations to restore the world to where it was before the sin. Once restored (fulfilled in the world to come), the mission becomes obsolete and the obligations are cancelled.
Every Jew performs his part in the Tikkun of the world in different aspects, according to his innate strengths and destiny either consciously or not. The basic assumption to explaining the Torah according to this approach, is that our forefathers were deeply aware of those missions and their personal designation, and did their best to resolve them totally or, at least, lessen the remainder for the rest of the times. Even if we don't really understand how the Tikkun works, they did it; and each of their deeds was meant to fulfill a certain mission.
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Therefore, by working 7 years for Lavan, Yaakov intended to fulfill a certain mission he was destined (more on that in another question), i.e "to free the world of some kind of the Evil of Lavan". He did it three times (7+7+6 years) overall.
The Numbers: many ideas in Judaism are divided into a certain number of levels against the spiritual significance of that number. Most famous are the numbers 10 (Sefirot), divided into 7 (lower) and 3 (higher), and 4 (letters of the Holy Hame). Each of them represents some kind of unity of subject "on all levels". For example, the 12 tribes (3 times 4) and the 12 hours and months and , as having 3 aspects (see Sefirot) and sub dividing into 4 (against the Holy Name) on each aspect.
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For example a week is divided into 7 days where each day represents a certain aspect (of Sefirot), hence the names Sun-day, Moon-day and up to Saturn-day. Although the work performed on each day is almost identical to another (with exception of Shabbos), the intentions (כוונות) are different, hence the Song of the Day.
When a Tzaddik performs certain actions a certain number of times, he intends to complete a mission on all levels. For example, Yaakov bowed to Eisov 7 times, having different intentions each time, to remove Eisov's wickedness and it finally worked. Another example would be the 8 days of Mileh, to ensure that the newborn will get through all the types of days (7) before the Mileh for the circumcision to be successful.
After this preface, I can now understand, that after we know Yaakov worked 7 years (and having in mind that he was in a great hurry to raise the 12 tribes as he was pretty old (77?), and could use his fathers wealth to marry the sisters) we could suppose that he intended to lessen Lavan's Evil by performing certain yearly work 7 times (1 Shemitah). And that was needed also to be performed a total of 3 times - against Rochel, Leah and the cattle. Only when that was accomplished (I don't explain the missing 21st year) Yaakov could finally leave Lavan.
[1] This tradition I learned personally from R' Moshe R. Luria, a descendant of Ariz"l
