Gemara

88b (Mishnah): If a Chacham says that there are five Parshiyos in Tefilin, to add to what Chachamim explained, he is liable.

Question: Also Lulav has an explicit source in the Torah, Chachamim explained it, and if one adds other species to the four, he invalidates the Mitzvah!

Answer: If Ein Tzarich Eged (the species need not be tied), holding additional species at the same time has no consequence. If Tzarich Eged, the bundle (with the extra species) was already invalid (from when he tied it.)

Question: Also Tzitzis has an explicit source in Torah, it is explained by Chachamim, and if one adds additional strings he invalidates the Mitzvah!

Answer: If the Torah does not require even one knot (after inserting them through the corner), extra strings do not invalidate the Mitzvah. If the Torah requires tying the strings, it was invalid from the start (due to the extra strings)!

Question: If so, also adding to Tefilin does not disqualify a valid Mitzvah! If he made four boxes and put a fifth next to them, it does not invalidate the Mitzvah. If he made five boxes the beginning, it was invalid from the start!

Answer: (Really, he made four, and put a fifth next to them.) R. Zeira taught that if the outer boxes are not exposed to the air, the Mitzvah is invalid.

Sukah 31a (Beraisa): Just like we do not detract from the four Minim, we do not add to them.

35a: "Pri Etz Hadar" cannot refer to peppers, for if one took one, it would not be noticed. The Torah said to take one, not two or three!

36b: (Mishnah - R. Yehudah): We tie the Lulav only with its species.

(Rava): He holds that Lulav Tzarich Eged. If one would tie it with another species, this is five Minim.

(Mishnah - R. Yehudah): We tie the Lulav only with its species.

Eruvin 95a (Mishnah): If one finds Tefilin (on Shabbos in a field, where they are prone to be disgraced) he brings them in one pair at a time;

R. Gamliel says, he brings two pairs at a time.

96a: The first Tana holds that one transgresses Bal Tosif (adding to Mitzvos) without intent. R. Gamliel huolds that one transgresses only with intent.

Rishonim

Rif (Sukah 16b): Just like we do not detract from the four Minim, we do not add to them.

Rambam (Hilchos Lulav 7:7): One takes one Lulav, one Esrog, two Aravos and three Hadasim. One may add to the Hadasim so it will be a big bundle. One may not add to or detract from the other Minim. If he added or detracted from them, he (some texts - did not) disqualify the Mitzvah.

Rebuttal (Ra'avad): We never heard that an addition disqualifies, nor of an Isur to add or detract, even according to R. Yehudah, who says that the Lulav must be tied. Rava said that R. Yehudah holds that one must tie it only with its species, for to use another species would be Bal Tosif. This implies that the same species, e.g, date shoots, is not Bal Tosif, even though there is more than one Lulav. All the more so there is no Bal Tosif according to Chachamim, who say that it need not be tied, and each stands by itself.

Magid Mishneh: All agree that one may add Hadasim, for they are to beautify it. Even though we do not require Eged, it is a Mitzvah to tie them for beauty. The Ra'avad holds that Bal Tosif is only with another species. We find that one who wears two pairs of Tefilin transgresses. This is difficult for the Ra'avad! His proof from date shoots is no proof. One who took mere date shoots does not fulfill the Mitzvah, therefore, Bal Tosif does not apply to them. They are not Lulavim! To resolve the Sugyos, I say that if he took two Lulavim at once, similar to Tefilin, he transgressed. If he took a different species, since we hold that Ein Tzarich Eged, he did not transgress, for each stands by itself. Still, l'Chatchilah we do not add, like it says in Sukah, and like the Rambam said (Halachah 5). There, he did not say that adding disqualifies the Mitzvah. One may add to beautify the Mitzvah, even the same species, even Kosher Hadasim. Some Ge'onim permitted adding Aravos for beauty. The Rambam (Teshuvah 96) said that since the Gemara mentions a Heter only for extra Hadasim, we should not add other species.

Rashba (1:75): Whatever is for beauty is not Bal Tosif. The Rambam holds that extra Hadasim beauty it, but Aravos do not.

Rosh (Sukah 3:14): One may add extra of the four Minim, and Bal Tosif does not apply, like it says in Rosh Hashanah. The Rambam permits adding only Hadasim. Any other addition disqualifies. The opposite is more reasonable! It says "Anaf Etz Avos" to teach that (exactly) three are required! "Arvei" teaches at least two, but also more is called Arvei! Chachmei Provincia questioned this. He retracted and permitted all. It seems that one may add even a Lulav or Esrog, even though it says Kapas (singular) and Pri Etz, for they are not different species. The Gemara said 'the Torah said to take one, not two or three', i.e. only one is required. Gedolim argue about whether Hadas that is not Avos (covered with leaves) is another species. Bahag disqualifies Hadas Shoteh (three branches do not emerge from the same junctures) with Kosher Hadasim. Rav Natrunai Gaon says that if one cannot find three Hadasim Avos, he fills the lack with Hadasim that are not Avos, and it is Kosher. Rebbi Paltiya says that they are different species. Even if they are Pasul for the Mitzvah, they grow in one patch, and they are one species, so Bal Tosif does not apply. This is like we say that according to R. Yehudah, we tie the Lulav with date shoots, but not with another species.

Poskim

Shulchan Aruch (651:14): One may not add another species to the four Minim, due to Bal Tosif.

Beis Yosef (DH u'Mah): The Tur says that this is only if he takes it l'Shem Mitzvah. In Rosh Hashanah, we learned that at the time of the Mitzvah, one transgresses even without intent!

Magen Avraham (27): R. Yerucham said that this (at the time of the Mitzvah, one transgresses even without intent) is according to Rava, who holds that Mitzvah do not require intent. We rule (OC 60:4) that Mitzvos require intent, so one transgressed only with intent. The Magid Mishneh and Ran hold that one does not really transgress, for the extra species is separate, but one should not do so, for he looks like one who adds. Perhaps this is the intent of the Tur and Beis Yosef.

Shulchan Aruch (15): One may not take more than one Lulav and one Esrog, but he may add extra Hadasim and Aravos if he wants. Some forbid to add a Hadas Shoteh (three branches do not emerge from the same junctures) to the three properly covered with leaves. Some permit. Meticulous people do not add to the two Aravos and three Hadasim.

Shulchan Aruch (664:7): Some say that one may not take the Aravah bundle on Hoshanah Rabah together with the Lulav when he is Yotzei (the four Minim), until after he blesses, takes and waves the four Minim. If he took the Aravah beforehand, he transgressed Bal Tosif. I say that this is wrong, but there is no loss to be concerned for this opinion. All agree that later, one should take the Aravah by itself to show that it is an obligation.

Taz (4): It is a mistake, for we hold that Lulav Tzarich Eged only to beautify the Mitzvah. Therefore, the Aravah bundle is like the Aravah of the Lulav (and one may have extra). Alternatively, it is a mistake because he does not intend to add to the Mitzvah, rather, for another Mitzvah.

Gra (7): The Magid Mishneh (Hilchos Lulav 7:7) says that for another species, Bal Tosif is only mid'Rabanan. L'Chatchilah it is forbidden. L'Chatchilah one may add another of the same species that is not Kosher for the Mitzvah (e.g. date shoots). If the same species is Kosher for the Mitzvah, Bal Tosif applies, e.g. two pairs of Tefilin, or throwing blood on more corners of the Mizbe'ach (Zevachim 80b). Whatever is to beautify the Mitzvah is permitted l'Chatchilah, whether it is the same or a different species. However, the Ge'onim forbid another species even for beauty. One may tie the four Minim with another species, for this is not the way it grows (facing up), and in any case the Ge'onim forbid only l'Chatchilah.

Mishnah Berurah (22): We take it afterwards, for after the Mitzvah one transgressed Bal Tosif only if he intends to add.

Mishnah Berurah (23): The Shulchan Aruch disagrees, for he permits taking extra Hadasim and Aravos.

Rema: The custom is to take the Aravah bundle with the Lulav during the wavings and Hakafos. It is better not to take it with the Lulav at all. Even one who takes them together should hold only the Aravos during the supplications for water.

Yom Tru'ah (Rosh Hashanah 28b DH Omar): If in Eretz Yisrael one was warned not to sit in a Sukah on Shemini Atzeres due to Bal Tosif, and transgressed, obviously he is lashed! This is why Chachamim obligated to invalidate the Sukah! If we discuss one in Chutz la'Aretz, surely he is not truly lashed; perhaps 'lashed' means only that he transgresses. However, since he does so due to Safek, surely he does not transgress at all!

Yom Tru'ah (Rosh Hashanah 28b DH Mesiv): The case of Birkas Kohanim teaches that Bal Tosif applies even to mere Dibur. However, since he intends to add his own Berachah, surely he intends to transgress. Why is this difficult only for Rava, who says that Mitzvos do not need intent? Also, he does not say that the addition is mid'Oraisa. The Rambam says that one transgresses only if he says that the addition is mid'Oraisa! It seems that he said so only regarding matters added for a fence. In general (e.g. Hilchos Tefilah 14:12), he agrees that intent to add suffices to transgress.

Maharal Diskin (Re'eh 13:1 DH Es): If one did something only because he thought it is a Mitzvah, does he transgress Bal Tosif? The Rambam says that if Beis Din ruled to make a new Mitzvah, they transgress Bal Tosif. If the people transgress, Beis Din would also transgress Lifnei Iver for making the people transgress Bal Tosif!

Note: It seems that this is unlike the Rashba (Rosh Hashanah 16a DH Lamah), who says that Bal Tosif is only what a person decides on his own, but not enactments of Chachamim.

Birkei Yosef (Sof Siman 243): The Ramban questioned why Rashi cited Beraisos that imply that the Torah forbids Amira l'Nochri (telling a Nochri to do Melachah on Shabbos). The Mizrachi answered that if people think that it is mid'Oraisa, this is even better (they will be more careful not to do it)! Some ask that according to the Rambam, this is Bal Tosif! The Ramach answered that the Rambam said so only regarding matters like meat of Chayos with milk, lest one think that one who transgresses is lashed mid'Oraisa. We are not concerned for Amira l'Nochri, for it has no action (so one is not lashed for it).