And the donkey saw me and turned aside before me these three times; had she not turned aside from me, I would now have killed you and let her live.
Midrash Tanchuma Balak 9 — three times corresponding to the three pilgrimage festivals.
English: “These three times [regalim]” — he hinted to him: You seek to uproot a nation that celebrates three pilgrimage festivals a year?
Wordplay on רגלים linking the three turns to the three festivals.
The angel’s three blockages are themselves a message: the people you would curse is defined by three annual ascents to the Temple.