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In general, one can pray an extra shmoneh esrei beyond the regular shacharit-mincha-maariv if one wants to do so. This is called a tefillat nedava -- a voluntary prayer.

I know that one cannot do this on Shabbat because no voluntary offerings were offered in the Temple on Shabbat due to restrictions on labor. However, on Yom Tov it seems that voluntary offerings could be brought (see Mishna Chagigah 1:4). As such, can one pray a tefillat nedava on Yom Tov?

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There is a dispute in Beitzah 20b whether voluntary offerings can be brought on Yom Tov, but the final halachah is that they cannot; the only private korbanos that can be brought on that day are the ones in which one is obligated for Yom Tov - the olas re'iyah, shalmei chagigah and shalmei simchah (Rambam, Hil. Chagigah 1:8).

[The mishnah you quoted means that you can bring them on Chol Hamoed (see Rambam ibid. 1:10), or that if you brought such a korban on Erev Yom Tov you can fulfill the obligation of simchah by eating the meat on Yom Tov (ibid. 2:11-12).]

Because of this, Rambam (Hil. Tefillah 1:10) equates Shabbos and Yom Tov when forbidding a voluntary prayer, and Shulchan Aruch (Orach Chaim 107:1) follows suit.

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