Sayyid Ibn Tawus and Shaykh al-Saduq have both narrated the following on the authority of Jabir ibn 'Abdullah al-Ansariy:

I visited the Messenger of Allah (a.s) on the last Friday of Ramadhan. As he saw me, he said, "Jabir: This is the last Friday of Ramadhan. You should thus bid it farewell by saying the following:

اللّهُمّ لا تَجْعَلهُ آخِرَ العَهْدِ مِنْ صِيَامِنَا إِيَّاهُ

allahumma la taj'alhu akhira al'ahdi min siyamina iyyahu

O Allah: (please) do not make it the last of our fasting in this month;

فَإِنْ جَعَلْتَهُ فَاجْعَلْنِي مَرْحُوماً وَلا تَجْعَلْنِي مَحْرُوماً

fa-in ja'altahu faj'alny marhuman wa la taj'alny mahruman

but if You decide so, then (please) make me enjoy (Your) mercy and do not make me deprived (of it).

Verily, anyone who says this supplication will win one of the two good things; either attaining the next Ramadhan or having the forgiveness and mercy of Almighty Allah."

Reference: Mafatihul Jinan