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No Ammonite or Moabite shall be admitted into the congregation of the Lord; none of their descendants, even in the tenth generation-Deuteronomy 23:4

What is the significance of the ten-generation number that the torah uses in this verse? What is it about ten in this context that makes it significant?

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With regards to a mamzer (Devarim 23:3), see Maayano Shel Torah, quoting a teaching from HaRav HaKadosh of Ostrovska (perhaps the Zichron Shmuel) :

A loose translation: Why does the Torah need to emphasize that a Mamzer is forbidden even after 10 generations?

There is an opinion in the Talmud Yerushalmi that a berya, a complete creature can become nullified if it is intermingled among a mixture of more that nine hundred and sixty.

If you calculate the percentage of a Mamzer that is left after 10 generations, it is 1/1024 - [Note that a Mamzer can marry a convert or descendant of converts. The child is still a Mamzer. Rambam, Hilchot Issurei Biah 15:7-8]

The child of a Mamzer and a non-Mamzerite woman would have 50% Mamzerness. [Halachically the child is completely a Mamzer, but biologically he is only half a Mamzer]

A grandson would have twenty-five percent from the father.

The third generation will have an eighth.

The fourth will have a sixteenth.

The fifth will have one thirty-second from the original mamzer.

The sixth will have one sixty-fourth.

The seventh generation will only have one out of one hundred and twenty-eight.

The eighth will have one out of two hundred and fifty-six.

The ninth will have one out of five hundred and twelve.

The tenth generation will only have one out of one thousand and twenty-four from the original mamzer.

Because one might argue that according to the minority opinion in the Talmud Yerushalmi the child my no longer be a Mamzer after 10 generations, the Torah explicitly refutes that claim.

I don't see why the same logic can't be used to explain why the Torah mentions 10 generations by a Moavite or Ammonite.

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But normally "tenth generation" would include the original mamzer, so the tenth would be 1/512 of a mamzer... – Ypnypn Dec 22 '15 at 2:05
"Halachically the child is completely a Mamzer, but biologically he is only half a Mamzer" – Menachem Dec 22 '15 at 3:20
My point is that the "10th-generation => 1/1024" logic seems flawed. – Ypnypn Dec 22 '15 at 4:54

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