What's added to damage when throwing a two-handed weapon, the thrower's Strength modifier or the thrower's Strength modifier ×1.5?
No matter how many hands a typical creature uses to throw the thrown weapon, the typical creature only adds its Strength bonus to damage
The Player's Handbook on Thrown Weapons says, "The wielder applies his or her Strength modifier to damage dealt by thrown weapons (except for splash weapons…)" (113).
Compare this with Two-handed that says, "Apply 1-1/2 times the character’s Strength bonus to damage rolls for melee attacks with such a weapon" (ibid. and emphasis mine).
An exception to this is, for example, the extraordinary ability thunderous throw of the prestige class bloodstorm blade that, in part, says
[Y]ou[, a level 2 or higher bloodstorm blade,] can apply 1-1/2 times your Strength bonus to damage if you wield the thrown weapon with two hands, and you can use Power Attack with your thrown weapon attacks (adding two times the number subtracted from attack rolls as a bonus on damage rolls when throwing a two-handed weapon). (Tome of Battle 101)
Nonetheless, an exception like this is necessary for such a damage increase.