Unless I am missing something, you can not make a bull rush as an attack of opportunity. A bull rush is either a standard action or a replacement for an attack during a charge. As opposed to Disarm and Trip, which can be made in place of any melee attack (which includes attacks of opportunity.)
Assume I could for sake of argument. How far would the person travel?
If my beard were made of green spinach, how much iron would it contain?
Let's say I charge and bull rush. I have greater bull rush so my bull rush makes the target provoke. I shield bash as my attack of oppurtunity. I have shield slam. Shield slam gives me a free bull rush on a successful shield bash. I bull rush.
Greater Bull Rush doesn't let you make an attack of opportunity. It lets your allies do so.
Benefit: You receive a +2 bonus on checks made to bull rush a foe. This bonus stacks with the bonus granted by Improved Bull Rush. Whenever you bull rush an opponent, his movement provokes attacks of opportunity from all of your allies (but not you).
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The AoO bull rush result takes precedence because it interrupts yours. Your original bull rush merely causes a provocation but you can't complete it because of the interruption.
So picture it like this you bull rush an orc on a charge causing the orc to become off balance and stumble back into the path of your shield swinging ally who swats him. The off balance orc then pitches in the direction of the shield slam's bull rush and stops the appropriate distance away.
--School of Vrock
Considering Shield Slam specifically lets you bull rush after each hit with a shield bash, along with other possible ways to bull rush on an AoO like the Push monster/eidolon attack (technically not bull rush, but basically same end effect), it can certainly come up...
I would say the distances would just stack with each other. PF doesn't require you to move with the target to rush them more than 5 ft like in 3E, so it's apparently all about ho much force you can shove them away with. Another bull rusher's effort would add to yours. Since both of you needing to beat enemy CMD and then by 5 more for each additional 5 ft is a high access barrier, straight up adding the bull rush distances together seems fine, and definitely the simplest way of handling it.
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I think they will not stack.
You ally shield slam don't interrupt your bull rush as it isn't the bull rush that provoke, but the movement of a target after a successful bull rush by a attacker with Greater Bull Rush. The target will move 5' or more away from you.
When your ally make his shield slam attack and his free bull rush attack he change the direction of the target movement, as your can't be on the same axis as the movement you imparted on the target (you would have bull rushed your ally, too). As the new movement is in a different direction, stacking them would generate some very weird move, so the simplest [Edited away on Lincoln Hill suggestion: best] solution is to have the new movement supersede the previous movement.
Yeah, my entire post could've just been summarized as "vector addition," I just kind of blanked on that.
It is very easy, and seems the fairest way to handle it. Although maybe it'd be better to use Pythagorean theorem for the hypotenuse between the two vectors. *shrug* When distance is done in whole 5 ft increments, perhaps that level of detail is best left out...
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No, you don't move your target the full movement from the first bull rush in one direction and then the full movement from the second shield bash in another direction. Doing it that way you generate a false set of movements.
The only way to do that that try to respect how adding vectors work is the use of "the Pythagorean theorem for the hypotenuse between the two vectors" as StreamOfTheSky suggested, but then you would have the problem to adapt that to a square grid.
You can get an approximation of that finding the final point of the creature movement sung the method that Driver 325 suggested and then moving the target there along a straight line and not along two different movement paths, but again you have the problem of the square grid. Almost certainly your target would still move in a strange way unless the two vector are at 90° of each other. Someone would have to choose in which square he would move at each step of his movement, and if that is a good way to generate infinite discussions at the table.
The shield slam on a charge+ramming shield idea is questionable. It could work only because the ramming ability is badly written and don't say that you have to take your swift action bull rush immediately after hitting with the shield slam during a charge.
Sequence of events:
1) charge,
2) shield bash when you reach the target
3) bull rush from your Shield Slam feat
4) the target move (the movement is a direct effect of your attack and happens as soon as you hit the target)
5) you can take your bonus shield bash from the ramming ability as a swift action but first you must resolve the previous action
6) bull rush say "You can move with the target if you wish but you must have the available movement to do so." so, if you have enough movement you can follow the target.
7) when the target stop, thanks to the hole in the ramming ability description, if you are in range, you can use the swift action bull rush granted by the shield.
But even with that hole, you aren't adding the effect of two bull rushes together, you are resolving two separated bull rush.
Assume I could for sake of argument. How far would the person travel?If my beard were made of green spinach, how much iron would it contain?
What do you mean, "if"?
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Shield Slam (Combat)
In the right position, your shield can be used to send opponents flying.
Prerequisites: Improved Shield Bash, Shield Proficiency, Two-Weapon Fighting, base attack bonus +6.
Benefit: Any opponents hit by your shield bash are also hit with a free bull rush attack, substituting your attack roll for the combat maneuver check (see Combat). This bull rush does not provoke an attack of opportunity. Opponents who cannot move back due to a wall or other surface are knocked prone after moving the maximum possible distance. You may choose to move with your target if you are able to take a 5-foot step or to spend an action to move this turn.
Read the feat. The bull rush get from the shield bash isn't a free action [it is free, not a free action]. It isn't even an action. It is part of the shield bash action you do, it even use the same attack roll.
So you can't take it later, after the swift action, you take it as part of the shield bash.When you try this kind tricks read the rules carefully.