(R. Yochanan): If Minchah and Musaf were in front of him (the time for Minchah arrived, and he did not yet pray either), he prays Minchah first.

Berachos 27a (Mishnah): The time for Musaf is the entire day;

R. Yehudah says, it is until seven hours of the day.

(Beraisa): If on Minchah and Musaf were in front of him, he prays Minchah first, for it is more frequent;

R. Yehudah says, he prays Musaf first, for its time will expire soon, but Minchah's time will not.

Rambam (Hilchos Tefilah 3:11): If one needed to pray Minchah and Musaf, he prays Minchah first, and afterwards Musaf. Some say that we do not pray Minchah first b'Tzibur, lest they err.

R. Mano'ach: The concern is lest they not pray Musaf afterwards. Alternatively, next year they will pray Minchah before Musaf, even if it is before the time for Minchah. The Gemara discusses an individual ('if Minchah and Musaf were in front of him'... he prays...')

R. Yonah (Berachos 19a DH Rebbi and d v'Yesh): On Yom Kipur, we pray Musaf after six and a half hours. We should pray Minchah first! Tosfos says that our Gemara discusses Minchah Ketanah, which is after nine a half hours of the day. In the time of Minchah Gedolah, since there is still much time in the day, he prays Musaf and afterwards Minchah. They support this from the Yerushalmi. This is wrong. We must say that our Gemara discusses Minchah Gedolah, for R. Yehudah says that one prays Musaf first, and he holds that Musaf is only until seven hours! How could he say that after nine and a half hours, he prays Musaf first? Its time passed! Rather, they discuss Minchah Gedolah. R. Yehudah says that Musaf is first, for its time will pass. My Rebbi explains the custom regarding Yom Kipur. Our Gemara discusses only one who wants to pray both Tefilos now. Therefore, he prays Minchah first. If he wants to pray only Musaf now, and to pray Minchah an hour or two later, he may pray Musaf first, even if the time for Minchah Gedolah came. On Yom Kipur, we do not intend to pray Minchah and Musaf next to each other. Since we intend to pray Minchah afterwards, we may pray Musaf first. However, once the time for Minchah Ketanah came, he prays Minchah first, like the Yerushalmi.

Rosh (Berachos 4:8): The text says that one prays Minchah first, like it says in Zevachim. Therefore, if Shacharis extends past seven hours on Yom Kipur, Minchah must come before Musaf. The Yerushalmi says that Minchah comes first only even if there is not time for both of them. Minchah is first only if the primary time for Minchah, i.e. Minchah Ketanah came. We are not concerned (lest he miss Minchah) if it is only time for Minchah Gedolah, i.e. seven hours of the day. However, the Bavli disagrees. Even regarding seven hours, it rules like Chachamim. Some say that 'there were two Tefilos in front of him' means that he needs to pray both now, e.g. he wants to start a big meal. He must pray Minchah before eating, so he prays Minchah first, for it is Tadir (more frequent) and this is its primary time. If he need not pray Minchah now, he may pray Musaf first, and delay Minchah until close to twilight.

Rashba (28a DH Tanu): Presumably, if he prayed Musaf first, he was Yotzei, even if he was obligated to pray Minchah first. He need not pray Musaf again, for its time is the entire day. Just l'Chatchilah, one must pray Minchah first.

Shulchan Aruch (OC 286:4): If one needed to pray Minchah and Musaf, e.g. he delayed praying Musaf until after six (Rema - and a half) hours, which is the time for Minchah, he must pray Minchah first, and afterwards Musaf.

Magen Avraham (1): Even though the Shulchan Aruch (233:1) says that l'Chatchilah one must pray Minchah Ketanah, since now he must pray Musaf, he must pray Minchah first, for it is the time. This is like one who wants to eat (once Minchah Gedolah came; he must pray first, so he may pray Minchah Gedolah). Nowadays we rely on people who call out when it is time to pray in the Beis ha'Keneses, so he need not pray Minchah now.

Bi'ur Halachah: If one delayed until almost seven hours, the Pri Megadim was unsure if all would agree that it is better to pray Musaf first, lest he transgress praying Musaf in the proper time.

Magen Avraham (3): It seems that if it is close to evening and there is not enough time to pray both, he prays Musaf, for there is compensation for Minchah at Ma'ariv.

Rebuttal (Beis Meir and Dagul me'Revavah): This is logical. However, the Rosh brought the v. Yerushalmi, which says oppositely!

Defense (R. Akiva Eiger): Perhaps the Yerushalmi discusses when compensation is impossible, e.g. his relative died, and after Shabbos he will be an Onen (who does not pray), for compensation is only during the next time for one of the three daily Tefilos. Alternatively, it discusses one who intentionally delayed praying until there is no time for both. There is no compensation, for he is considered Mezid. The Magen Avraham did not discuss such a case.

Rema: If he prayed Musaf first, he was Yotzei.

Gra: What is Tadir comes first, i.e. l'Chatchilah.

Shulchan Aruch (ibid): Some say that this is only if he needs to pray both now, e.g. he wants to eat, and one may not eat before Minchah. However, if he need not pray Minchah now, he may pray Musaf first.

Mishnah Berurah (12): The Rema (232:2) concluded that only a big meal is forbidden before Minchah Gedolah, and a Shabbos Seudah is not called a big meal. (Only Seudas Bris Milah or Nisu'in is called big.) Therefore, we are not concerned lest he not pray, especially nowadays that people call out when it is time to pray in the Beis ha'Keneses, so he need not pray Minchah first.

Kaf ha'Chayim (34): Even according to the opinion that permits a big meal before Minchah Gedolah, here we discuss one who does not want to be lenient (to eat before Minchah). Alternatively, since he needs to pray Minchah and Musaf, all forbid a big meal beforehand, for there is more reason to be concerned lest the meal get drawn out.

Kaf ha'Chayim (35): The first opinion (Tosfos) does not rely on this, for he requires praying Minchah first even on Yom Kipur.

Tur: My father (the Rosh) used to pray Musaf alone on Yom Kipur, in order to pray in time.

Beis Yosef (620 DH v'Tov): Hagahos Maimoniyos (Sof Shevisas Asor) says that the Shali'ach Tzibur must ensure that Shacharis not extend until six and a half hours, for then some rule that one must pray Minchah before Musaf. For this reason, Maharam used to rush Shacharis greatly, and sometimes skip Avinu Malkeinu when needed.

Bach (3): Surely, every Rav must warn people to finish Shacharis and not delay with Nigunim, so Musaf will begin before six and a half hours, to avoid the Machlokes. If six and a half hours already came, we rely on this opinion and pray Musaf first, since on Yom Kipur one need not pray Minchah (right away). Also, we rely on the Rambam, who says that we do not pray Minchah first b'Tzibur, lest they err. This is even if the time for Minchah Ketanah came.

Rema: However, if the time for Minchah Ketanah came, he prays Minchah first.

Shulchan Aruch (ibid): Some say that we do not pray Minchah first b'Tzibur, lest they err.

Mishnah Berurah (14): The concern is lest they pray Minchah before Musaf even before midday.

Kaf ha'Chayim (36): This opinion, of the Kesef Mishneh, holds that an individual must always pray Minchah first. The Ari Zal says that Musaf says that based on Kabalah, Musaf must be before Minchah, even for an individual, unless he wants to eat before Musaf. (I.e. since it is already the time for Minchah, to eat before Musaf he must pray Minchah first - PF.)