According to Jacobus Arminius's definition of free will, did Pharaoh have free will? Through the first chapters of Exodus up to the literal exodus, God kept telling Moses that he would harden Pharaoh's heart. Does this mean that Pharaoh didn't have (Jacobus Arminius definition of) free will?
It depends on what one means by “free will.”

If, by free will, one means, a person (in this case, Pharaoh) can make free choices ACCORDING to his nature, then yes, Pharaoh had free will. He had a sinful, fallen nature, so he was able to freely act in accordance with that nature—to sin and practice evil. 

All who are born in Adam have a fallen nature, free to be what they are: sinners. But no one has the freedom within that fallen nature to not sin, nor to obey God, because their desire is for sin and their mind is corrupt. It requires a NEW NATURE to be able to obey God, and only God can grant that new nature to a person, according to the good pleasure of His sovereign will, to the praise of His glorious grace.

Man fell (in his nature) when Adam sinned, and this act of rebellion was decreed by God from all eternity, even though Adam was truly free to either obey God or disobey God, as in his original state, he did not have a fallen nature and could do what he wanted without the propensity to sin, unlike ourselves (and Pharaoh).

When it says that God hardened Pharaoh’s heart, it doesn’t mean Pharaoh acted against his will; it means that he acted according to his already fallen nature, which was hardened, and so God carried out His will through Pharaoh, to not free the Hebrews upon Moses’ plea, until such a time that God could display His great power and glory against Egypt, which was a type and shadow of bondage to sin (exactly what the sinful nature is). Ultimately, God is control of all things, and if God had freely chosen to show mercy to Pharaoh, he would have. But His purpose for Pharaoh was different.

Romans 9:17-18


  17 For the Scripture says to Pharaoh, "For this very purpose I have
  raised you up, that I might show my power in you, and that my name
  might be proclaimed in all the earth." 
  18 So then he has mercy on whomever he wills, and he hardens whomever he wills.   


1689 London Baptist Confession of Faith - On Free Will 1