Gemara

Mishnah: If one did not bring Chagigah on the first day of the festival, he may bring it any time during the festival, even on Shemini Atzeres.

Chagigah 9a - Question: In what sense is the rest of the festival Tashlumim?

Answer #1 (R. Yochanan): Every day is Tashlumim for the first.

Answer #2 (R. Oshaya): All the days are Tashlumim for each other (the Mitzvah intrinsically applies on each day).

R. Zeira: They argue about one who was lame (and therefore exempt) on the first day, and healed on the second day. R. Yochanan holds that since he was exempt on the first day, he is exempt the second day; R. Oshaya obligates him once he heals.

Beitzah 20b - Beraisa - Beis Shamai: The time for Olas Re'iyah is not fixed!

(Mishnah): If one did not bring Chagigah on the first day of the festival, he may bring it any time during the festival, even on Shemini Atzeres.

Beis Hillel say, the time is fixed!

(Mishnah): If one did not bring Chagigah during the festival, he is exempt from bringing it afterwards.

Rishonim

Rambam (Hilchos Chagigah 1:1): If one entered the Azarah on the first day of the festival and did not bring Olas Re'iyah, he transgressed the Lav of "V'Lo Yera'u Fonai Reikam". He is not lashed for he did not do an action.

Rambam (ibid. 4): If one did not bring Olas Re'iyah or Shalmei Chagigah on the first day of the festival, he may bring it any time during the festival - "Shiv'as Yomim Tachog la'Shem" (Devarim 16:15) teaches that every day is fit for Chagigah, they are all Tashlumim for the first day.

Question (Kesef Mishneh): Why does the Rambam rules like R. Yochanan? R. Oshaya was greater than him!

Answer (Lechem Mishneh): R. Oshaya who codified the Tosefta was greater than R. Yochanan. If the Gemara cited him, his opinion would precede R. Yochanan's. Rather, this R. Oshaya was an Amora. R. Yochanan was greater than him, so the Halachah follows R. Yochanan. Also, Chagigah 9b shows how R. Yochanan could hold like the opinion that Pesach Sheni is an independent festival; this suggests that the Halachah follows R. Yochanan.

Rambam (2:5): If one was lame or blind on the first day, or became Tamei the first night he is exempt from Chagigah and Re'iyah. However, if he became Tamei the first day he must bring them during the festival when he becomes Tahor.

Lechem Mishneh (1) and Ri Korkus (4): The Rambam holds that there is Tashlumim for both of these. We learn this about Shalmei Chagigah from the Mishnah (Chagigah 9a); we learn about Olas Re'iyah from Beitzah 20b.

Imrei Baruch (on Turei Even Chagigah 4a b'Emtza Devarav): The Lechem Mishneh says that the Rambam agrees with the SMaG, that one does not transgress "V'Lo Yera'u Fonai Reikam" as long as he brings the Korban during the Regel. This is not so! The Rambam holds that one who entered the Azarah empty on the first day transgressed. There is Tashlumim for Olas Re'iyah, but this does not fix the Lav.

Shirei Korban (Chagigah 3b DH Hifrish): The Rambam holds that if one entered but did not offer Olas Re'iyah there is no Tashlumim.

Rebuttal (of Rambam - Turei Even Chagigah 2a DH Le'asuyei): The Rambam only discusses Tashlumim for Chagigah. He holds that Re'iyas Panim applies only on the first day; since the Chiyuv for Olas Re'iyah comes from "V'Lo Yera'u Fonai Reikam", Tashlumim for Olas Re'iyah is only for one who was obligated on the first day. The argument about whether or not the other days are Tashlumim for the first is only regarding Chagigah. This is difficult, for the Gemara (9a) does not require separate Drashos for Chagigah and Olas Re'iyah - surely we learn one from the other! A suggestion is made that 'Re'iyah l'Chetza'im' means bringing some of one's children to the Azarah on one day, and the rest on another day; no one objected to the idea of Re'iyah after the first day. Rather, we learn Re'iyas Panim from Chagigah; the same argument about Tashlumim for Chagigah applies to Re'iyas Panim.

Rebuttal #1 (of Shirei Korban and Turei Even - Or Some'ach 1:1): The Rambam holds that there is Tashlumim for Re'iyas Panim. He does say (2:6) that the Isur "V'Lo Yera'u Fonai Reikam" applies only on the first day, i.e. the first day of the Regel that he enters the Azarah.

Rebuttal: #2 (Sefas Emes Chagigah 2a DH Le'asuyei): All agree that we learn Olas Re'iyah from Chagigah. However, the opinion that the other days are Tashlumim for the first says that we learn only about Olas Re'iyah, but Re'iyas Panim is only on the first day. The opinion that says that each day is an independent obligation holds that there is Tashlumim for Re'iyas Panim.

Rebuttal #3 (Minchas Chinuch 489-490 DH v'Hinei Kosavnu): The Rambam holds that the Ikar Re'iyas Panim is the first day; if one did not enter on the first day, it is the first day he enters. On the Ikar day one is liable each time he enters without a Korban, even if he brought a Korban earlier in the day! On other days he is liable only if he did not yet bring a Korban.

Aruch ha'Shulchan ha'Asid (Chagigah 196:12,13,15): "Es Chag ha'Matzos Tishmor...V'Lo Yera'u Fonai Reikam" suggests that the Lav is only if he did not bring the Korban the entire Regel. Further, if there is no Tashlumim for Olas Re'iyah, it would have a fixed time and it would override Shabbos even though it is not a Korban Tzibur, just like Pesach! The Rambam holds that one may bring Olas Re'iyah only on the first day of the Regel that he enters the Azarah.

Tosfos ha'Rid (Yerushalmi Chagigah 2b DH ha'Kol): The Rambam learns from the Yerushalmi that the Mitzvah of Re'iyas Panim is only on the first day. Nevertheless, we learn from Chagigah that there is Tashlumim for Olas Re'iyah.

Tosfos (Chagigah 6b DH Yesh): The Tosefta (1:4) teaches that there is Tashlumim for Simchah, but not for Chagigah and Olas Re'iyah. Our Gemara did not bring it because it holds that all of these have Tashlumim.

Rebuttal (Ri Korkus): One is obligated to eat Shalmei Simchah every day of the festival. If so, there cannot be Tashlumim! Our text of the Tosefta says that Simchah applies all seven days, but Chagigah and Olas Re'iyah do not.

Note: Sefas Emes (Yoma 3a b'Tosfos) says that the Rambam (Chagigah 1:1,3) holds that the Chiyuv of Simchah is to eat once during the festival. It seems that this is the opinion of the Tosfos in Chagigah (see also Tosfos Chagigah 7b DH Olos) even though elsewhere Tosfos holds like the Ri Korkus (Pesach 95b DH Ta'un, Yoma 3a DH Pazer, Rosh Hashanah 50 Amos DH Mah and Sukah 47a DH Linah). Ben Aryeh alters the text of Tosfos to cite the Tosefta like we have it, and to say that the Gemara did not bring the Tosefta because the Gemara holds that also Chagigah and Olas Re'iyah apply all seven days.

Rambam (ibid. 7): Shemini Atzeres is included in Tashlumim for Sukos. Tradition teaches that there are six days of Tashlumim after Shavu'os, just like for the first day of Pesach.

Ritva (Sukah 47b DH uv'Tosfos): According to R. Yochanan, the days of Sukos are Tashlumim for one who was obligated on the first day; Shemini Atzeres is a Regel by itself to obligate one who was exempt on the first day. (It does not obligate one who brought Chagigah - there are three Regalim, not four. According to R. Oshaya, every day is Tashlumim even for one who was exempt on the first day; nevertheless, it is necessary to teach that Shemini Atzeres is Tashlumim, for the verse discusses only the seven days of Sukos.