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/)
30 “But as for thee and thy servants, I know that ye will not yet fear the LORD God. 31 For although the flax and the barley were smitten (for the barley was in the ear, and the flax was in bloom) 32 but the wheat and the spelt were not smitten (for they ripen late).”
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.
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:
It was the heart which was weighed against the feather of truth in the hall of Ma'at during the diving judgement of the deceased. A heart unburdened with the weight of sin and corruption would balance with the feather and its possessor would enjoy the eternal afterlife.
The vital importance of the heart in determining the fate of the deceased in the afterlife lead to a chapter in the Book of the Dead (Spell 30) where the deceased implores his heart not to betray him. In part, it reads:
"O my heart which I had from my mother, O my heart which I had upon earth, do not rise up against me as a witness in the presence of the Lord of Things; do not speak against me concerning what I have done, do not bring up anything against me in the presence of the Great God, Lord of the West."
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.
And I will harden Pharaoh’s heart, and multiply my signs and my wonders in the land of Egypt
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:
But though he had done so many miracles before them, yet they believed not on him: That the saying of Esaias the prophet might be fulfilled, which he spake, Lord, who hath believed our report? and to whom hath the arm of the Lord been revealed? Therefore they could not believe, because that Esaias said again, He hath blinded their eyes, and hardened their heart; that they should not see with their eyes, nor understand with their heart, and be converted, and I should heal them.
Which references Isaiah 6:10:
Make the heart of this people fat, And make their ears heavy, and shut their eyes; Lest they see with their eyes, and hear with their ears, And understand with their heart, and convert, and be healed.