My Rabbi mentioned the famous gemara regarding King Menashe in shul today:
והיו מלאכי השרת מסתמין את החלונות שלא תעלה תפילתו של מנשה לפני הקב"ה והיו מלאכי השרת אומרים לפני הקב"ה רבונו של עולם אדם שעבד ע"ז והעמיד צלם בהיכל אתה מקבלו בתשובה אמר להן אם איני מקבלו בתשובה הרי אני נועל את הדלת בפני כל בעלי תשובה. מה עשה לו הקב"ה חתר לו חתירה מתחת כסא הכבוד שלו ושמע תחינתו הדא היא דכתיב ויתפלל אליו ויעתר לו וישמע תחינתו וישיבהו
This is from Talmud Yerushalmi Sanhedrin 10:2.
When Menashe wished to repent, the Malachei HaShares (Ministering Angels) wished to hide his prayer from God so they closed all the windows to Heaven. God therefore decided to bore a hole beneath His Throne of glory in order to allow Menashe's prayer to enter, so much does He desire teshuva, true repentance.
The imagery struck me as familiar. It reminds me of the story of Og & Noah as brought down in Pirkei D'Rabbi Eliezer, Chapter 23.
How was Og saved from the Mabul (Flood?) He told Noah that he would be a servant to him and his children forever if only Noah would save him. Thus, he lay on top of the ark and Noah drilled a hole into its ceiling. Noah would pass Og's food to him through this little hole in the Ark and during the times that he was not receiving food, Og stopped up the hole with his little finger. (For a beautiful retelling of the story check out Howard Schwartz's The Diamond Tree.)
I think it is deliberate that the imagery echoes. When Noah saved Og, he drilled a little hole in the Ark through which to pass up his food to him. In return, Og saved Noah and everyone in the Ark by ensuring the rainwater could not get into the Ark through the hole and stopping it up with his little finger. When God wished to receive Menashe's prayer, thereby saving him from the conditions he was in (he was being boiled in a pot at the time), he chose to bore a hole underneath His Throne of Glory in order to receive his prayer.
Sometimes God is pleased by man's innovations. It occurred to me that perhaps God saw Noah's compassion for Og, for whom he chose to bore a little hole in the Ark so that he could pass him his food, and thought to Himself: If mortal man can take compassion upon a giant and bore a hole within the Ark for him, how much the more so can I take compassion upon my creatures and bore a hole beneath my throne for them! And thus does one man's kindness eventually benefit us all.