With a full party, you will strike 6(six) times.
In a Double Battle, if a Pokémon uses Beat Up on its partner, the partner will attack itself (unless it suffers from a status ailment).
In a double or triple battle, if a Pokemon uses Beat Up on one of its teammates, then that teammate will attack itself.
Tested on PO (Thank you B3N for all your help):
My Weavile used Beat Up on My Terrakion. He lost 22 HP collectively.
But, when I saw his HP, it was 75 when it should have been 78. Those last 3 points must be from him hurting himself. I used a calculator for all the math to be sure.