Gemara

89a (Mishnah): Three and a half Lugim were used for the Menorah, half a Log for each lamp.

(Beraisa): "Me'Erev Ad Boker" - we put enough oil in to burn from evening until morning;

Chachamim estimated that a half-Log suffices (for the longest nights, i.e. in winter).

Shabbos 21b (Beraisa): The Mitzvah is for Ner Chanukah to burn from sunset until the last people cease from the market.

Suggestion: If it extinguished, one must relight it!

Rejection #1: No, this teaches that if one did not light at sunset, he may still light until this time.

Rejection #2: No, this teaches that one must put enough oil to burn until this time.

Rishonim

Rif and Rosh (Shabbos 9a and 2:3): The last people leave the market half an hour after Shki'ah.

Rambam (Hilchos Chanukah 4:5): If one forgot or was Mezid and did not light at the end of Shki'ah, he lights until the last people cease from the market. This is half an hour or more (after Shki'ah). One must put enough oil to burn until this time.

Rosh: If one did not light at sunset, he may still light until the last people leave the market. After this, the time ended. It seems that we, who light indoors, and there is a Heker (it is noticeable) only to the household, need not be so particular about this. Alternatively, one must put enough oil to burn until this time. Alternatively, after this time one may extinguish it or use its light.

Tosfos (89a DH v'Shi'aru): We are not concerned if oil remained in summer. The Yerushalmi says that in summer they used thick wicks (to use up the full Shi'ur of oil), and in winter they used thin wicks. Initially, they used medium wicks.

Mizrachi (Shmos 27:21 DH v'Shi'aru): The Shi'ur of oil depends on the thickness of the wick! Perhaps the Neros of the Menorah had a container for the wicks, and this obligated the wicks to be a fixed thickness. There is no Shi'ur to the wicks for Ner Chanukah, so Chachamim did not fix a Shi'ur of oil. The fixed a Shi'ur only for how long they must burn.

Shiltei ha'Giborim (Shabbos 9a:6): Half a Log (the volume of three eggs) of oil burns the entire night, like we find regarding the Menorah, so an eighth of a Log burns half an hour. If one put a minimal amount of oil, and it occurred that it burned past half an hour, this requires investigation whether one may extinguish it.

Note: This assumes that the night is 12 hours. Half a Log was the Shi'ur for the long winter nights. If they would light the Menorah 18 (or 36) minutes after sunset, (and dawn is 90 or 72 minutes before sunrise), in the longest nights in Eretz Yisrael, this is almost exactly 12 hours before dawn. However, some Meforshim say that we light the Menorah before Shki'ah. One opinion (Pesachim 59a) holds that we light it before Korban Pesach!

Rebuttal (Magen Avraham 3): The amount of oil needed depends on the thickness of the wick! (It is not fixed.)

Defense (Machatzis ha'Shekel DH l'Fi): Regarding multiple wicks in one bowl of oil, Shiltei ha'Giborim (10a) says 'they must be the same thickness, according to the Shi'ur of oil for Ner Chanukah, i.e. an eighth of an egg.' I.e. he holds that the wicks in the Mikdash had a uniform thickness, and wicks for Ner Chanukah should be the same. This is unlike Tosfos Yom Tov (and Tosfos above).

Note: Shiltei ha'Giborim could hold like Tosfos, just he holds that the wick for Ner Chanukah must be the minimum thickness used in the Mikdash, i.e. in winter.

Panim Yafos (Shemos 27:1): The wicks for the Menorah should not be too thin, for we require "Leha'alos Ner" (to illuminate). They should not be too thick, for then it looks like a torch. Chachamim gauged the proper thickness, and found that half a Log was enough for winter nights. It is proper to make wicks for Ner Chanukah similar to the Mikdash. Meforshim ask what was the miracle on the first day (the oil would normally burn for one day). We can say that there was enough oil to burn the entire night only with a very thin wick, and they saw that the oil increased and filled a half Log in each Ner.

Note: This connotes they expected to use thin wicks, but after they saw that the Neros filled up with the usual Shi'ur of oil, they used proper wicks. Why would the Kohen Gadol seal a flask with a Hechsher, if the oil inside was too small for the proper Mitzvah? Perhaps it was intended for a single Ner, and they used it for all seven.

Beis She'arim (OC 28 DH Aval): If oil (e.g. of Tamei Terumah) must be burned, we do not say that it is not the proper Shi'ur for Ner Chanukah, since the wicks can be any thickness. The Shi'ur is only how long they must burn.

Maharil (28): If one did not light Ner Chanukah one night, there is no compensation that night. He lights normally on other nights. This is not like the Omer, in which one who did not count one night lost the Mitzvah. There, it says "Temimos" (full weeks). Here, each night is a Mitzvah by itself, just like Lulav (every day is a separate Mitzvah). This that we add (a Ner every night does not show that it is a continuation of the previous night;) it is merely Hidur (beautication of the) Mitzvah. One who ate garlic and gives off a bad smell (should not eat more garlic. One who neglected the Mitzvah one night should not neglect it another night.)

Poskim

Shulchan Aruch (OC 672:1): Some say that if one is busy, he may light earlier, from Plag ha'Minchah (one and a quarter hours before the end of the day), as long as he puts enough oil to burn until the last people cease from the market.

Darchei Moshe (1): Semak brings from R. Shimshon that even nowadays one must make long wicks. Others say that nowadays one may take small Neros that will burn for a short time. Even so, it is improper to take less than the normal Shi'ur.

Magen Avraham (3): Even nowadays one must put in this amount of oil, unlike thise who are lenient.

Kaf ha'Chayim (15): The Darchei Moshe held that one should not be lenient. In the Rema, he did not mention the lenient opinion. If one put in too little, Chayei Adam says to extinguish it and relight without a Berachah.

Bi'ur Halachah (DH ka'Zeh): If one does not have the proper Shi'ur, he lights without a Berachah.

Kaf ha'Chayim (17): If one has a Shi'ur for one Ner, since this is the primary Mitzvah, he lights it and blesses.

Darchei Moshe (1): I say that even R. Shimshon only meant to argue with the latter opinion, but he did not require wicks longer than the Shi'ur. Some people use Neros that burn until midnight. This is needless. It is not Hidur Mitzvah, for after they burned the required time, one may benefit from them. Hidur Mitzvah applies only at the time of the Mitzvah. Neros Chanukah of the Beis ha'Keneses are different, for the Mitzvah is for them to burn the entire day.

Magen Avraham (3): There is no Hidur Mitzvah to put extra oil in a Ner. If one burns wax candles, if they are longer they are nicer, like Neros Yom Kipur, and Lulav and Esrog (it is Hidur Mitzvah if they are bigger than the minimal Shi'ur).

Mishnah Berurah (6): If one uses wax candles, it is Hidur Mitzvah if they are long. In any case they should not be too long.

Kaf ha'Chayim (18): Eliyahu Rabah says that since we hold that one may not use them as long as they burn, perhaps there is a Hidur Mitzvah even for an oil Ner that burns longer.

Magen Avraham (2): The Shulchan Aruch connotes that if one lit after Shki'ah, it suffices to put in enough oil to last until the last people cease from the market. The Agudah says that the Rambam connotes that it must burn for more than the time to walk a Mil.

Mishnah Berurah (5): This was in the days of the Gemara, when the Heker was for people passing by outside. Nowadays, the Heker is only for the household, so we require the normal Shi'ur, even if one lights after the last people left the market.

Mishnah Berurah (4): If one lit early but put in oil only for half an hour, he adds oil, but he does not bless again.

Kaf ha'Chayim (12): Eshel Avraham says that if on Erev Shabbos one put only enough oil for a half an hour, perhaps he is Yotzei even though it will not burn at night. We find that if it extinguished before night he need not relight it, unlike during the week. However, perhaps since one may light from Plag ha'Minchah in pressed circumstances (even on a weekday), oil for half an hour was enough, so perhaps he would not bless when relighting it. Chayei Adam says that if it does not burn at night, he was not Yotzei, so he would bless again. Since Poskim argue, if one did not put in enough oil, he extinguishes it and lights again (with enough oil), but he does not bless again.

Kaf ha'Chayim (13): Surely, one must put in a Shi'ur of oil before lighting it.

Shulchan Aruch (2): If one forgot or was Mezid and did not light at Shki'ah, he lights until the last people cease from the market, which is about half an hour. Then people pass by, and there is publicity of the miracle.

Beis Yosef (DH v'Chasuv b'Orchos): Orchos Chayim says that a case occurred in Lunil in which someone lit only two Neros on the third night, and they were stringent that he light what he lacked. He need not bless again, for his first Berachah was for his obligation for all the Neros.

Mishnah Berurah (6): This is even if he intended to light only one Ner when he blessed, and later he obtained another Ner. He does not bless again because the Chiyuv is only one Ner, and the rest is merely Hidur Mitzvah.

Shulchan Aruch (2): If one did not light the entire night, there is no compensation.

Rema: On other nights he lights like everyone else, even though he did not light a previous night.

Kaf ha'Chayim (29): A case occurred in which one did not light one night. He heard the ruling of the Shulchan Aruch, and he was pained. Ru'ach Chayim saw this, and told him that there is a Tikun to add oil like the Shi'ur for two nights.