If your Brash Taunter would fight your own Obosh, the Preypiercer, Brash Taunter would deal 12 damage to your opponent.
Fighting means two creatures deal damage equal to their power to each other:
701.12a A spell or ability may instruct a creature to fight another creature or it may instruct two creatures to fight each other. Each of those creatures deals damage equal to its power to the other creature.
Obosh doubles all damage from sources with odd CMC. Obosh is such a source, so it deals 3 damage to Brash Taunter, doubled to 6. Taking damage triggers Brash Taunter, so it deals 6 damage to a target opponent, again doubled to 12.
Obosh's doubling ability is a replacement effect, which can only apply once to any given event.
614.5. A replacement effect doesn’t invoke itself repeatedly; it gets only one opportunity to affect an event or any modified events that may replace that event.
However, the damage from Obosh to Taunter, and from Taunter to opponent are two separate events that can both be replaced by Obosh's ability, so you end up with quadruple the total damage to your opponent.