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?