Continuous fillet.
If you're talking about Filleting to a non-zero radius and the Lines' ends meet, you can JOIN or PEDIT/Join them into a Polyline [or more than one], then use Fillet with the Polyline option and it will Fillet all the corners at once. Even if they don't meet but are sort of close, PEDIT/Multiple/Join has a fuzz factor, and the same can be used to Fillet them all to zero radius [Explode the resulting Polyline if you want them to remain as Lines].
What you describe could be done, but has a potential error situation, again with non-zero radius, in that the place where you picked on a given Line in one Fillet may be part of what gets lopped off by it, and if such a routine functions by remembering your pick points [it's hard to imagine that it could do otherwise], it will run into a problem on the next Fillet, with nothing there to pick. But if you're talking about zero radius, or can live with the occasional failure, it should be doable.
Kent Cooper, AIA 