Both sides used Greek fire, which may have had some explosive properties…no one is really quite sure what Greek fire was made of anymore so it's hard to be sure. It was, at the very least, some kind of inextinguishable fire.

But there was one instance during the Siege of Acre where this random guy showed up and showed Saladin’s troops how to fling even stronger Greek fire at the crusader siege towers. He gave them a demonstration, destroyed three towers, then disappeared. The ultimate source for this story is Saladin’s secretary, Imad ad-Din al-Isfahani, but since that’s not available in English, here is the story from the latter 13th-century historian Ibn al-Athir:

“…a man from Damascus was passionate about collecting equipment used by Greek fire specialists and acquiring ingredients that strengthen the action of fire. People who knew him used to blame him for this and disapprove of it but he would reply, ‘This is not a practice I engage in myself but I just want to understand it.’ It was for God’s own purposes that he was in Acre. When he saw the towers erected to attack Acre, he began to make the highly combustible substances that he knew of, such that no clay, vinegar or anything else would resist. When he had finished, he came before Emir Qaraqush, who was in charge of affairs in Acre and the governor there, and said to him, ‘Would you order the artificer to launch what I shall give him from the trebuchet facing one of these towers to set fire to it.’ Now Qaraqush’s exasperation and his fear for the city and its inhabitants were almost killing him. These words added to his exasperation. He turned on him angrily and said, ‘These specialists have already done all they can in launching Greek fire and the like without success.’ Someone present said, ‘Perhaps God has arranged for our deliverance at this man’s hands. It will not harm us to consent to what he says.’ Qaraqush then agreed and ordered the artificer to do what he ordered. He launched several pots of oil and various substances without having ignited them. When the Franks saw that no pot set fire to anything, they yelled, danced and cavorted on the roof of the tower. Eventually, when he ascertained that what he had thrown had hit the tower, he shot a full pot, having ignited it. The tower was set aflame. He shot a second and a third and the fire blazed out in various parts of the tower and was too quick for the men on its five storeys to flee and save their lives. When they had seen the first pots have no effect, their arrogance had made them feel secure and not bother to run for safety, so that God gave them an early taste of fire in this world before the next. After the first tower was burnt, he moved to the second, from which the occupants had now fled in fear. He set that one on fire and likewise the third. It was a day to remember, the like of which had not been seen. The Muslims [outside] watched in delight, their faces lit up after their despair, in their joy at this victory and the Muslims’ escape from slaughter, because there was not one amongst them who did not have either a relative or a friend in the city. This individual was taken to Saladin who offered him large sums of money and a substantial fief. However, he accepted not a single bean, saying, ‘I did it just for God Almighty. From Him alone do I wish for reward.’” (vol. 2, pg. 373-374)

I suppose the developers of Assassin’s Creed might have had this story in mind. Unfortunately it’s not really clear if the pots exploded, or if they were somehow just even more powerful Greek fire. It would be another few decades at least before gunpowder was introduced to the Middle East, so they certainly didn’t have anything like that yet.

Sources:

Jonathan Phillips, The Life and Legend of the Sultan Saladin (Yale University Press, 2019)

M.C. Lyons and D. E. P. Jackson, Saladin: The Politics of the Holy War (Cambridge University Press, 1982)

Donald S. Richards, trans., The Chronicle of Ibn al-Athīr for the Crusading Period from al-Kāmil fī'l-ta'rīkh, part 2 (Ashgate, 2007)