Who knows seventy-seven?
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77 are the number of contemporary tzaddikim whose merit Moshe invoked to save the Jewish People after the sin of the Golden Calf: Aharon, his four sons, Pinchas, Calev, and the seventy elders. Hashem demanded the merits of eighty tzaddikim, though, and so Moshe filled the bill by invoking also the three Patriarchs. (Devarim Rabbah 3:15)
Incidentally, in another midrash (Shemos Rabbah 44:7) the seventy elders are omitted, and Moshe speaks only of seven contemporary tzaddikim plus the Patriarchs. There is a story in which R' Shneur Zalman of Liadi, questioned on this point by one of his opponents, resolved it by referring to a dispute elsewhere about whether each tribe is considered a separate "community" (in which case each one needs the merits of ten tzaddikim) or not (and then ten suffice for the entire nation).
SEVENTY SEVEN are the number of days in a year on which one may not get married (in chutz laaretz).(not including Shabbos- see comment).
2 days of Rosh Hashana
1 day Yom Kippur
9 days of Sukkos
8 days of Pesach
33 days of Sefiras Haomer (According to most opinions only 33 days are forbidden. Those who keep more are doing so because of the different opinions when the 33 days take place-see relevant comment below).
2 days of Shvuos
22 days between 17 of Tamuz and Tishah B'Av (inclusive).