RAW: Only one opportunity to use LAs when going solo versus solo. However, if the monster has a way to bring in allies...
RAI: I don't think they were worried about this issue at all. I do not believe they had any intentions relating to what should happen when a solo PC faces off against a legendary monster.
RAF: This is likely not going to be a fun encounter if this matters. There are few legendary encounters that are a fair fight for a solo PC, even if the PC is 20th level. If the monster is trying to kill the PC, with or without the legendary actions, they'll likely win.