109. If one wishes to daven an extra Sh'mona Esrei (called a tefillas "nedavah") one may do so at any time, so long as a chiddush (novelty - ie. a new personal bakasha) is added to the tefillah. L'chatchila, one must add a chiddush in middle of each bracha of Shmona Esrei - except for the first 3 brochos and last 3 brochos where one should not add anything. However, Bi'dieved it is sufficient if one added only one single new bakasha in one bracha.
Shulchan Aruch with Mishna Brurah 107:1.2
110. On Shabbos and Yom Tov one may not daven a tefillas "nedavah" because one may not bring a korbon "nedavah" on Shabbos or Yom Tov; and, one may not daven a Mussaf as a tefillas "nedavah" because a korbon Mussaf is never brought by an individual, only by a tzibbur.
Shulchan Aruch with Mishna Brurah 107:1
111. If one is unsure whether he davened Sh'mona Esrei (for Shachris, Mincha or Maariv) he may daven another Sh'mona Esrei and make the following condition: "If I have not davened this should be my obligatory tefillah, and if I already davened then this should be a tefillas nedavah. {Under these circumstances no chiddush must be added to the Sh'mona Esrei because the resolution of the sofek is itself the chiddush in the tefillah.}
Shulchan Aruch with Mishna Brurah 107:1
112. One may not use the method above for a sofek about a Sh'mona Esrei of Shabbos, Yom Tov or any Mussaf. The only alternative to resolve the sofek and still daven in the proper z'man is for the person to act as a Shliach tzibbur for chazoras Hashatz for the Sh'mona Esrei in question. Shulchan Aruch with Mishna Brurah 107:1 (see Biur Halacha 124:1)
113. If one began a Sh'mona Esrei because he was sure that he had not yet davened, and then while still in middle of Shmona Esrei realized that in fact he had already davened, he must stop where he is, and he may not continue this Sh'mona Esrei as a tefillas "nedavah" even if he now wants to add a chiddush to it. {Reason: There is no korbon which is part chov and part nedeva; and this tefillah was mistakenly started as a chov}.
Shulchan Aruch with Mishna Brurah 107:1