75a - We learned in a Braisa: a ger until 10 generations is allowed to marry a mamzeres; from then on he is prohibited to marry a mamzeres. And some say until the non-Jewish identity is removed from him.
Rambam (Isurei Biah 15:8-9) paskens like the latter opinion, that the son of ger and giores can marry a mamzeres (the offspring will be mamzerim) and (Isurei Biah 19:12) cannot marry a kohen. Rambam rejects 10 generations and holds once either the mother or father is a yisrael, even though one parent is a ger, that child may marry a kohen (IB 19:12 & 16, from Rebi Eliezer Ben Yaakov in the mishna on 77a) and is forbidden to marry a mamzer/mamzeres, because one yisrael parent removes the non-Jewish identity. Ran (30b in Rif, s.v. umutar) (also see Chelkas Mechokek & Beis Shmuel on S"A 4:22) asks on Rambam that if a ger marries a bas yisrael, their daughter should be forbidden to marry a kohen, because the mishna on 66b says if there is kiddushin and no aveirah the vlad follows the father, which is a ger, who is not allowed to a kohen? A simple answer to the Ran's question is, we pasken khal geirim is not a khal, but there is a khal for kohanim, leviim and yisraelim. When the mishna says follow the father, that's when the father has a khal (or a psul). If the mother is a yisraelis, that's enough to put the child into the khal. But if the father is a ger, he has no khal, so he can't pass on nothing to his child. If the mother is also a giores, the child would have the same status of a ger and be asur to marry a kohen. (Note that the hava amina in Bechoros 47 holds that a son of a goy and a bas levi would be a levi, because where there is no mishpachas av the child would get mishpachas eim.)
The Rosh paskens like the first lashon in the Gemara, the we do follow the 10 generation bit. See how the Shulchan Aruch brings this halacha in 4:22.
Reb Chaim (Issurei Biah 15:9) is about this Ran & Rambam. His answer is more complex, and I direct interested readers there.
(It's interesting - see Aruch Hashulchan 4:11 and 7:39 - ger biyamim kadmonim - he says it that way because it was prohibited from the dina d'malchusa (the Czar) to convert to Judaism. See his lashon in his title to Y"D 268: "Laws of converts in the early days, and in our countries we have no authority to accept converts due to the law of the land." Has anyone written on the historical aspects of conversion? We know that the Gra accepted R' Avraham ben Avraham, the Ger Tzedek of Vilna (the former Polish count who converted), but that was 100 years before the Aruch Hashulchan, and probably a different country also, (but I'm not well versed in 18th and 19th century Eastern European history either).