Free will is axiomatic to Judaism. Being able to make the choice between right and wrong is one of the features that makes us human. Without free will, life would either be futile or even senseless. That is why I find it so disheartening to read multiple times in the Exodus narrative of the hardening of Pharaoh's heart (Exodus 4:21, 7:3, 7:13, 9:12, 10:1, 10:20, 10:27, 11:10, 14:4, 14:8, 14:17). If G-d hardens one's heart, He interferes in one's emotions and motivations in a way that one would not feel otherwise. Looking at the text, this interference didn't help Pharaoh "see the light." Reading "[and] I will harden the Pharaoh's heart" (ואני אקשה את לב פרעה) literally means that Pharaoh's free will has either been distorted or nullified.
As Dr. David Shatz at Yeshiva University points out, there are three huge problems with the "literal view." One, how can anybody hold the Pharaoh responsible if G-d removed his free will? Two, if Pharaoh has no free will, how can Pharaoh perform teshuva? If one were to use this as a justification for the Pharaoh, then we could all theoretically use that to abscond personal responsibility (Midrash Rabbah, Exodus 13:3). Three, if Pharaoh was not responsible, would that mean that G-d effectively caused evil? As an additional question, why was this even necessary? Couldn't G-d have taken care of the problem "with the snap of His fingers?" To think of G-d as an enabler or perpetrator of evil, חס ושלום, is perturbing. So how do we resolve the hardening of Pharaoh's heart with the notion of free will?
Maimonides thought that the negation was justifiable [as a punishment] because Pharaoh previously used his free will to cause such evil (Hilchot Teshuvah 6:3) that one cannot arguably repent for it. Even if the Pharaoh were that irredeemable, taking away his free will would be, at best, superfluous. Shimon ben Lakish realized a shift in language between the first five plagues and the second set. In the first five plagues, Pharaoh was the one hardening his own heart. In the second set, G-d did the hardening, which is to say that G-d was acting "measure for measure." In spite of Pharaoh's evil, it still does not answer how G-d would completely negate one's free will, and it makes the whole "giving Pharaoh a choice" a farce. Also, R. Moshe David Cassuto noted that since all actions are attributed to G-d, there is no difference between the first and second sets.
G-d's hardening of the heart did not negate the Pharaoh's free will, but rather it provided Pharaoh with free will. What would have happened if G-d decided not to harden Pharaoh's heart? Pharaoh most likely would have been so overwhelmed by G-d's awesomeness that Pharaoh's instinctive reaction would have been to let the Jewish people go. This interpretation means that the hardening of Pharaoh's heart was the way to ensure that the Pharaoh could withstand the plagues (Sadiah Gaon) in order to be able to choose between good and evil (Sforno; Josef Caro). Sforno quoted Ezekiel 18:23 in the process, thereby reminding us that no one is beyond repenting.
I like the idea that G-d's intervention meant preserving Pharaoh's free will. It certainly beats the option of free will deprivation as a punishment. However, my issue here is that I believe in an impersonal G-d, which is why I like R. Abraham ben Izra's interpretation. According to ben Izra, one does not interpret the phrase literally. G-d allowed Pharaoh to exercise his free will and follow his hardened heart. Rabbi Samuel David Luzzatto also went with a non-literal interpretation by saying that "the hardened heart" is a hyperbolic literary device alluding to Pharaoh's stubbornness. To go off of these two non-literaral interpretations, I would conjecture that "the Pharaoh's hardening of the heart" meant that Pharaoh was suffering from cognitive dissonance. There are moments in life where we see only what we want to see and ignore everything else. The Pharaoh was no exception to the common, human phenomenon.
In conclusion, G-d did not remove the Pharaoh's free will. How could He? Judaism has such respect for free will that, as the rabbinic commentary illustrates, G-d could not possibly have done so. The lesson I would take from the hardening of the Pharaoh's heart learning from the Pharaoh's mistake. We cannot let life knock us down and harden us to the point where we are incapable of doing the right thing. To pursue a life of goodness and looking on the bright side provide the antidote for us to avoid the Pharaoh's fate.