When we say shelo asani goy in brachos, we're thanking Hashem for not making us a goy. But what about geirim...are they supposed to say this bracha? Also, what about someone who did a geirus lechumra?
The question of whether or not converts recite the blessing goes back many centuries.
Some hold that he does recite the blessing. (Cf. Ba'er Heitev (OH 46)).
Don't Say it
נשאל הרמ"ה אם יכול לברך גר שלא עשני גוי. ושבוי שלא עשני עבד והשיב: כך ראינו שאין הברכה אלא על תחלת ברייתו של אדם לפיכך אין הגר מברך שלא עשני גוי
R. Abulafia responded that he does not recite the blessing. This is also the opinion of R. Isaiah Horowitz and R. Joel Sirkis (cited in Magen Avrahan (OH 46:4)), as well as R. Ben-Tsion Abba Shaul (Shu"t Or Letsion II:2:2).
Say a variant
Rama (OH 46:4) indicates (cf. Magen Avrahan there) that a convert recites a separate blessing of שעשני גר; who made me a convert.
Recite the blessing without God's name
The Kaf HaHayyim and R. Jacob Emden (cited in Tehumin (15, p. 444) write that a convert should recite the normal blessing, of שלא עשני גוי, but without God's name. In this vein, R. Yitshak Yossef writes in Yalkut Yossef that a convert shouldn't recite the normal blessing, but he may recite it without God's name:
ילקוט יוסף תפילה ב נוסחאות התפלה והברכות
יז. גר צדק אינו מברך "שלא עשני גוי", ואם ירצה יברך בלא שם ומלכות
It seems likely that the principle of safek berakhot lehakel of avoiding blessings that one may not be required to recite, would apply to one who converted out of doubt, and he would therefore avoid saying the blessing, at least with God's name. R. Ari Yitshak Shevat writes in an article in Tehumin (ibid) that the issue of those who converted as a stringency bring this issue to the fore. If I understand correctly, he means that if converts can't recite the blessing, then possible converts can't either.
It's a makhlokhet. According to Halachipedia, The Avudraham (cited by the Beit Yosef 46:4) says no because the b'racha is about how God created him. But the Shaarei Knesset HaGedola (cited by the Kaf Hachaim 46:36) says yes because the b'racha is thanks for not creating him to remain a non-Jew. Halachipedia further notes:
Magen Avraham 46:10 says that according to the Mekubalim a ger can recite Shelo Asani Goy because the bracha is referring to the state of one's Neshama when it will be taken from the world. Piskei Teshuvot 46:11 says one has on whom to rely if one wants to recite Shelo Asani Goy.
They don't say how the Avudraham, who says a ger does not recite this b'racha, rules for a ger l'chumra. The others say that a ger can recite, so if even a "regular" ger can then certainly a geir l'chumra can.