It is my understanding that Pharaoh truly believed that he was allowing the Jews out (even towards the very end) for a 3 day journey and the reason behind this journey was to go to Sinai in order that the Jews could perform sacrifices to their God. In addition, Pharaoh's understanding was that these sacrifices couldn't be performed in Egypt because the animals being sacrificed (Lamb) were a "God" of the Egyptians and it was too dangerous for the Jews to sacrifice in Egypt. Given all this, we know, from midrashim that Pharaoh was informed of the events of Shabbos Hagadol where the Jews were tying lamb to their bedposts in preparation for a sacrifice, which should have raised some red flags to him since he hadn't given a go-ahead to the Jews to leave. What is even more concerning to me though is why did Pharaoh allow the Jews to leave at all given that on the night of maakot bechorot when Pharaoh approached Moshe's residence, presumably, he saw ample evidence of a sacrifice being performed right there in Egypt! The easy answer is that he just let them out and sent soldiers with the Jews, because he just wanted the plagues to end and he would worry about it later, but this seems a little simplistic and I wanted to see if the MY crowd would be able to add some insight.
Par'o already suggested that they stay and bring karbanos in Egypt and Moshe answered him that they could not worship Hashem in a country full of idols. Indeed, when Par'o begged for the plague to end, Moshe answered that he had to go outside the city to ask for the plague to end. By the end of the ten plagues Par'o had panicked to such an extent that he was no longer thinking of excuses, he just wanted to get rid of them. That is why after he had a chance to work on himself he tried to go after the Bnai Yisrael and force them back.
The question is really, why didn't Pharaoh let the Jews out earlier? Any semi-rational person would have realized that, for whatever reason, the Jews' God wants them to leave the country for a while and is willing to turn over Egypt to get this done. It doesn't really matter if one understands why he wants them to leave, but clearly he does and Egypt is going to suffer if they can't leave. After only a few plagues, one should realize that conditions in his country aren't going to get any better, so he had better let the Jews do whatever they wanted if he wants his country to survive. And Pharaoh did admit that he was wrong in keeping them, and that Hashem was bringing these plagues upon him, but he didn't let them leave because Hashem "hardened the heart" of pharaoh so that he wouldn't let the Jews leave, and Hashem would be able to finish all of the plagues and show both the Egyptians and the Jews that he is in control of heaven and earth. After the Jews left Egypt, when Pharaoh again chased them into the sea, it was again because Hashem "hardened his heart" and compelled him to chase after the Jews even though it made no sense to do so.
So the final answer: Past a certain point, Hashem was controlling every action of Pharaoh so that His miracles would be apparent.