Pharoah starts by hardening his own heart, but then God hardens it (here is the list of verses). Did God remove Pharoah's free will?
That God removed or lessened Pharaoh’s free will is a common explanation; usually justified by saying that the plagues were punishment for the slavery and could not be allowed to be escaped. I never liked that explanation, but it’s out there.
Rabbi Samson Raphael Hirsch in his Commentary explains that God did not “harden Pharaoh’s heart” so much as “allow Pharaoh’s heart to be hardened”. This was achieved allowing Pharaoh to (incorrectly) perceive limits to God’s power in bringing the plagues.
For example, Hirsch translates Exodus 9:30–32 as a single quote, something like (adapting the JPS translation from http://mechon-mamre.org/)
I.e., Pharaoh is stricken with hail, but is allowed to believe that God is not actually powerful enough to destroy all of Egypt’s crops. The plagues get progressively harsher, but each time God leaves Pharaoh enough room to believe that “this is as bad as it will get”.
In the next chapter, Pharaoh’s servants recognize this tactic, demanding, “How long shall this man be a snare unto us?” The word “snare” is interesting, implying there was something “tricky” about the presentation of the plagues although they came with full warning.
קָשָׁה, Strongs 7185 means harden, and has the preformative attachment, אֶ, denoting the imperfect conjugation, used for incomplete action. This can be present or future depending on context, which in this case is obviously future, so we get "I will harden".
A man can only use his will within the limits of his nature. So the first point is that Pharoah being unregenerated could only make decisions within his sin nature; free will means that while man can make any decision, he will only make the decisions that his sin nature allows him to make.
The main point though is that God is in charge of everything. He doesn't remove man's free will, but he does direct his heart and actions to fulfill his eternal purpose established before the foundation of the world. He is sovereign.
Here is a New Testament record of God hardening hearts, John 12:37-40:
Which references Isaiah 6:10:
The Hebrew text uses three different words in this context:
I do not know if God's "hardening of Pharaoh's heart" is in conversation with the modern question of free will vs. determinism, but it is clear that the heavy/hard heart was an important and meaningful symbol within that historical and political context. See for example:
The question of whether or not Pharaoh had free will, like the question whether or not anyone has free will, is not addressed directly by the Pentateuch. God's repeated use of heart imagery to describe Pharaoh's insubordination is probably a satirical allusion to Egyptian mythology. God promised: "...I will bring judgment on all the gods of Egypt" (Exodus 12:12), so its no wonder that the language of Exodus is in conversation with Egyptian mythology. This article and this book, show how each of the Ten Plagues was intended as an affront to one of the many Egyptian gods.