If you realized that you didn't yet daven minchah and it's past shekiyah, when is the latest that you can begin praying the Amidah? I'm looking for different people's cutoffs.
YU Torah online has a good summary.
The subject is disputed by the Vilna Gaon and Rabbeinu Tam. The first allows only up till sunset; the second up to when the stars appear.
Mishna Berurah 233:14, limits the leniency to recite Mincha until tzeit hakochavim. He cites the opinion of P'ri Megadim, Eshel Avraham 233:7, who rules that one cannot actually recite Mincha until tzeit hakochavim, but rather until Rabbeinu Tam's shekiat hachama which is a few minutes before tzeit hakochavim. [R. Ovadia Yosef, Yechaveh Da'at 5:22, cites numerous Acharonim who disagree with P'ri Megadim and maintain that according to Rabbeinu Tam, one may recite Mincha until tzeit hakochavim.] Mishna Berurah then notes that even those who normally follow the opinion of Rabbeinu Tam should nevertheless show deference to the opinion of the Vilna Gaon and recite Mincha before astronomical sunset.
Nevertheless, Mishna Berurah, Sha'ar HaTziun 233:21, rules that even according to the Vilna Gaon, there is room for leniency in a pressing situation. R. Ovadia Yosef, op. cit., notes that although the Vilna Gaon himself does not allow one to recite Mincha after shekiat hachama, one can argue that within the opinion that shekiat hachama occurs at astronomical sunset (i.e. the Vilna Gaon's opinion) one can still maintain that latest time for Mincha is at tzeit hakochavim. According to R. Ovadia Yosef, the question of whether evening (for the purpose of Mncha) begins at shekiat hachama or tzeit hakochavim is not necessarily connected to the question of whether one follows Rabbeinu Tam or the Vilna Gaon.
One should be extremely careful to finish davening Mincha by Shkiah, sunset, since many early Poskim (12) hold that it is forbidden to daven Mincha after that time (13). It is better to daven on time without a Minyan than to daven after the proper time with a Minyan (14).
B'dieved, one may daven Mincha up to 20 minutes (15) after sunset. Some Poskim suggest that when davening Mincha this late, the following condition (Tnai) should be stated: If the present time is still "day", then my Tefillah is Mincha and my next Tefillah will be Maariv. If, however, the present time is already "night", then this Tefillah should be counted as Maariv and the next one will be Tashlumim (a makeup) for Mincha (16).
12 Rabbeinu Yona, Shiltei Giborim, Levush, Gra.
13 Mishnah Berurah 233:14; Aruch Hashulchan 233:9; Igros Moshe OC 1:24
14 Mishnah Berurah 233:14. Many other Poskim, however, hold that it is better to daven with a Minyan even if the Minyan will commence after sunset, see Mor Uketzia 233; Minchas Elozer 1:23; Einayim L'Mishpat Brachos 27a.
15 Mishnah Berurah allows one to daven Mincha B'dieved up to 15 minutes before the stars come out. Since, according to some views, the stars are out 35 minutes after sunset, the latest time Mincha may be davened is up to 20 minutes after sunset.
16 Biur Halacha 233:1.
The first Lubavitcher Rebbe says one can be lenient to pray Mincha during Bein Hashmoshos (twilight) which in Russia during the equinoxes it's about 30 minutes after sunset).
see here http://chabadlibrary.org/books/adhaz/piskey/17.htm (2nd to last paragraph)
In the mishna in Berachos 26a, the tana'im dispute until when one can daven mincha. R' Yehuda says until plag hamincha (the second half of the time of mincha ketana, which goes from 9 1/2 hours of the day till twelve), the Chachamim say until the evening. The gemara (26b-27a) proves that 'until plag hamincha' cannot mean until the end of plag hamincha (twelve hours), as then there would be no dispute between R' Yehuda and the Chachamim (as they also hold that one has until the evening, ie the end of twelve hours of the day). Thus it is clear that according to the most leninent opinion (the Chachamim), the time after which one cannot daven mincha is also the time that the hours of the day end.
It is well known that the Magen Avraham and the Gra argue about this, the M"A holds that the day goes from Alos HaShachar until Tzeis HaKochavim, whereas the Gra (and the Levush) holds that it is from sunrise till sunset (in truth this is debated by the rishonim as well).
It should also be pointed out that the view of the M"A is based on the position of Rabeinu Tam, that halachic sunset is 54 minutes after the viewed sunset (the Levush argues that even according to Rabeinu Tam the day goes from viewed sunrise to viewed sunset, but without Rabeinu Tam the view of the M"A cannot start).
The Levush also points out that those who held that the day goes from alos hashachar till tzeis mistakenly believed that at the equinox, day and night would each be twelve hours according to this. Thus reality proves that in fact day starts at viewed sunrise and ends at viewed sunset.
All those who allow davening mincha after shekia take the view of Rabbeinu Tam into account. Thus for most of us, who do not take Rabbeinu Tam into account even at the end of Shabbos, there is no justification for davening mincha after the viewed sunset. See Responsa Bemareh HaBazak, part 8 (http://eretzhemdah.org/Data/UploadedFiles/SitePages_File/123-sFileEn.pdf), siman 1.
one may daven mincha up until tseis. many are careful to do so earlier b/c otherwise when shabbos or yom tov is approaching one wouldn't daven minchah at the later time and so perhaps it's best not to daven minchah later in the day ever. even though one may daven minchah until tseis it is commonly accepted to do so before shkia so as to also be able to recite tachnun which is no longer said from shkia onward.
The latest time to pray tefilat mincha b'diavad is until tzet hakochavim; which is the time when three, medium-sized stars appear in the sky(see Brachot 26a, Pesachim 94a, and Shabbat 35b that states in the names of Shmuel, Rabi Yehuda, and Rabi Yose that it is still day until two medium-sized stars appear in the sky; and Shulchan Aruch OC 233:1 and Rama there; Yabia Omer 7:34; Yalkut Yosef 233:4; and Mishna Berura 233:7 & 14). Another calculation is about 13.5 minutes(b'sha'ot zmaniot) after sunset; the time estimated to walk 500 amot(SA ibid).
There are stricter opinions that limit the time of tefilat mincha, even b'diavad until sunset(like the Rambam, Talmidey Rabeinu Yona, Shiltei Hagiborim, Haknesset Hagedola, Lechem Chamudot, Halevush, Aruch Hashulchan, and the Vilna Gaon).
Due to the machloket Tanaim regarding the actual time of shkia, Chazal identified bein hashemashot as the time between sunset and tzet hakochavim as being "safek min hayom safek min halaila". However, the pashtut of the statements by Rabi Yehuda and Shmuel on daf 35b in Shabbat indicate that bein hashemashot is the time between when two and three medium-sized stars appear in the sky. In my opinion, it seems that this is the actual time of bein hashemashot, and Chazal extended the time to sunset because nobody can be exact with this shiur. In my opinion, since bein hashemashot is doubtfully day/night, it can be permitted to recite tefilat mincha during that time because the instutition regarding bein hashemashot at sunset is mi'd'rabanan and it is still not laila vadai(which seems like a reliance on the principle of safek d'rabanan l'kula).
Several years ago, when I was saying Kaddish for my father A"H, the shul where I was at for Mincha was waiting for the last men to arrive for the minyan. I noticed that it was getting pretty late and I mentioned to one of the rabbis that it was dark outside and it was almost too late to daven Mincha. He told me, "Don't look outside". I carry that bit of wisdom with me.