This might answer what the Gemara means in Chagigah 9b. The Gemara there says exactly what the Mishnah Berurah you cited says:
one who reviews his studies one hundred times is not comparable to one who reviews his studies one hundred and one times.
The Baal HaTurim explains the significance of the number 101 in his commentary on Shemos 32:34:
הנה מלאכי ילך לפניך מלאכי עולה בגי' מאה ואחד. ילך עולה כמנין הלכה רמז למה שאמרו אינו דומה מי ששונה פרקו ק' פעמים לשונה פרקו ק''א שעי''כ ילך ההלכה לפניו. גם הנה. עולה כמנין הלכה. גם ישקני מנשיקות ''פיהו'' עולה מאה ואחד. הנה מלאכי אותיות מיכאל שהוא מלאכי המיוחד לי:
The Baal HaTurim explains that the word מַלְאָכִ֖י - angel, has the numerical value of 101.
In the sefer Koveitz Chukai Chaim, a anthology of pratical halachic topics by Moreinu Horav Chaim Aharon Bleier shlit'a, the rav of the Cheder Horo'oh in Yerushalayim, the following is added to this topic:
Indeed, Chazal in Maseches Chagigoh indicate that there is something special about the number 101 when they tell us that "A person who reviews what whe learned 100 times does not compare to a person who reviews 101 times". The Baal Haturim, commenting on the posuk in Shemos 32:34 writes that מַלְאָכִ֖י in gematrio equals 101 and ילך equals הלכה, hinting to this Chazal - if one reviews 101 times, the halochoh will go before him, so to speak.
In fact, these two ideas are intertwined. The Chasam Sofer quotes the
Rama MiPano, who says that forgetting comes from the samech-mem, which
is 100 in gematrio. Reviewing 101 times defeats the samech-mem,
enabling a person to remember what he learned.
Source:
קובץ גליונות ליקוטי ופסקי הלכות חוקי חיים - Koveitz Chukai Chaim - see "The Significance of 101 Issues", p. 14