At the beginning of this week's parsha, Moshe mentions that he pleaded with HaShem to allow him to enter Eretz Yisroel but to no avail. The sefer M’galeh Amukos says that Moshe Rabbeinu davened 515 times - the gematria of Vaeschanan. R' Yehonasan Eybeschutz, in Divrei Yehonasan, is curious to discover how such a tally is reached.
He offers the following possibility: The Midrash states that Moshe Rabeinu started davening on 15 Av. As the gemara (Bava Basra 121a) explains, it was on this day that it was realized that the punishment for the sin of the spies was complete and no more men would die in the midbar. He saw that that decree had been fulfilled and had a glimmer of hope that perhaps, since he had been spared from the decree, he was in a position to plead for Divine Mercy. (This would explain why he never engaged in such extensive prayer on Aharon's behalf as Aharon died prior to 15 Av.)
There are 6 months from Elul to Shevat. We may assume that it was a normal year, whose months alternate between 29 and 30 days throughout. So those full months would total 177 days (3x30 + 3x29). Add the 16 days of Av that Moshe davened and the 7 days of Adar until he dies and we have 200 days. Of those 200 days, 28 are Shabbosos on which it is not permissible to make personal requests. That leaves 172 days. Considering Shacharis, Mincha and Maariv and we now have 172x3 = 516 tefilos. Only off by 1. However, the nation only discovered in the morning of the 15th of Av that the dying has stopped. Therefore, Moshe would have missed the Maariv from the night before and only begun davening at Shacharis. And there you have exactly 515 tefilos!
The Tur writes that on Yom Kippur one is permitted to make personal requests, but on Rosh HaShanah, Sukkos, or Shmini Atzeres it is forbidden. We would then have to subtract three more days of prayer. However, we are taught that Moshe Rabbeinu died on Shabbos. If that is the case, then Rosh HaShanah, Sukkos, and Shmini Atzeres of that year all fell on Shabbos as well. So we need not subtract for them and we are safe with our tally of 515!
Special thanks go out to R' Ari Storch for providing me with the material for this shtikle.