This used to be ruled that you could play as much as you wanted, and THEN Special Initiative kicked in,
if it could. For instance, if I played an end the battle card after using a different card to remove the hero, SI couldn't happen since the battle ended. You may see references in old threads to this sort of thing.
THIS HAS CHANGED.
Per the new rules on Special Initiative, it happens IMMEDIATELY when an ability would remove the last character from a side of the battle. Please review the new rule which goes into much more detail, but essentially removing the last hero grants SI even if you have Arrogance, which as pointed out essentially means that you do not pass initiative based on the numbers in the battle.