Barbican

Card Type: Holding
Gold Cost: 6
Keywords: Fortification
Card Text: Defending Personalities at this Province's battlefields have +1F.
Before comparing the strength of a Ranged Attack from a defending card at this Province's battlefield to its target, double its strength.
Legality: Clan Wars (Imperial), Four Winds (Gold), Hidden Emperor (Jade)
Deck: Dynasty

Printings

Erratum: Reads, "Barbican 6G Fortification. Each Personality gains +1F while defending this Province. Double the strength of each defending card's ranged attacks at this Province after applying all other Ranged Attack strength bonuses." [MRP, Gold Edition, 1 July 2001] Rules Archive, August 18, 2007, p. 78
Ranged Attack strength is doubled before applying Reactions that modify the Ranged Attack in any way besides increasing its strength (e.g., Stars Scatter), because traits precede actions. [JA, email, 11 December 2000] Rules Archive, August 18, 2007, p. 78
Clarified: Ranged Attack strength is doubled after all other bonuses. Text Change & Errata List, February 15, 2000, p. 12

Q: So with 3 Barbicans under Kaiu Sensei you could Double, Double, Stars Scatter and double again.
The answer you were looking for is a card effect only applies once from a particular copy of that card.
So once a particular Barbican doubles the original ranged attack no matter what other effects may occur to it that Barbican's doubling effect cannot apply again.
A: The counterargument to that is that, once Stars Scatter has been played, the new RA's are not the same RA as the original one. They are all completely new Ranged Attacks, and should get all existing bonuses again. For balance's sake, though, I think that the whole thing, Stars Scatter and all, should just be counted once when assessing bonuses. Vaughn Derderian, "Rules for December 11th", 11 Dec 2000