How would you get out of the guillotine that Shogun was in. I was thinking maybe he could reach around his back to untangle Chael's foot or anyone for that matter so he can prop himself up or turn to the side. I remember Ruas and Ebersole were always able to somehow get out of guillotines.
I'm not usually one to write anything up about the technical aspects of what some of the worlds best should do, so this is purely a subjective opinion and what I would coach someone to do, in terms of my right and wrong.
Personally I'd start with long before the guillotine got locked up, in fact if you watch the previous fight(Overeem v Browne) it illustrates what I'm about to talk about.
Shogun ended up in a front headlock on his knees with his hands on Chael's hips, and instead of addressing Chael's hands and breaking the hands apart or fighting the grip off of his neck, he simply stood up bent over at the waist. Which if you've done jiu jitsu for a couple months you'll know is spoon feeding a guillotine. If you watch Browne v Overeem, you'll see that at one point Overeem ended up with a similar front headlock, he appeared to be attempting a ten finger guillotine, however Browne still defended the way Shogun should have, he addressed the hands and broke Overeem's grip while still on his knees. Staying down doesn't offer your opponent as easy an ability to pull guard and close it up and finish.
Next once Chael pulled guard, it's difficult for me to see if Shogun's left arm was trapped under Chael's leg or if it was just Shogun's elbow and his hand was on the grip.
If Shogun's arm was trapped, he was in serious trouble and there is not a whole lot left to do, it is difficult to create enough force in a cross face where you have nothing to defend your neck. His best chance if that was the case would be to lay to his right hip and put create a difficult angle for Chael to finish the choke on. It would also keep Chael from getting his right hip to the mat and really putting significant pressure on Shogun's neck. Frankie Edgar did something similar to this when he fought Benson Henderson, I believe it was the first fight.
If Shogun's hand was not trapped and he was able to get some form of resistance on Chael's grip then he has more options. First he probably should not have posted up on his head, that does very little to stop a skilled guy from finishing a guillotine, he needed to flex his back and shrug his shoulders, making it difficult for Chael's arm to sink all the way under. Also posting on his head took away all of Shogun's ability to pull his head out, as he is use it as a point to support all of his weight. He should have once again created a difficult angle for Chael and put pressure on his right hip, rolling Chael onto his left hip and not allowing him to have full use to squeeze with his left arm (which in this case is not the arm doing the choking, but the arm applying the pressure required to finish the choke). Next Shogun needed to use his right arm to not just cross face Chael and turn his face, but to actually offset his neck/spine. Doing that takes away the majority of someone's ability to squeeze (Think attempting to throw a punch while looking straight up at the ceiling or lifting weights while looking at the ground). Shogun needed to use his right arm to force Chael's head backwards and to the side, which not only would/could take a large amount of the pressure away, it would also force Chael to extend his arms to a degree further weakening the hold.
Ideally, Shogun is able to get his right arm on the right side of Chael's face/head and is able to then start pulling Chael's head in a half circle around to the other side (Chael's left hip) which would relieve even more pressure. At which point the choke is pretty much off and it is just a matter of waiting for Chael to let go/burn his arms out or the round to end, or possibly breaking the hands apart and getting the head out.
There are a couple of different ideas in there, hopefully I didn't lose anyone, but not to avoid answering the question or fit into the old instructors cliche, but in actuality Chael was going to finish that 99/100 times, because Shogun stood straight up into it and didn't just prevent the situation and address it early. At least 90%, probably more, of great BJJ defense is prevention, never allowing someone to get that far into a hold.
I may have gone way deeper into that then expected, but I have a tendency to do that, I believe it was Killz a while ago who asked me something about my stretching routine and I posted damn near a book for him with pictures and all of kinds of stuff haha.