According to the simple meaning of the verses, there must have been at least three: the one whom Yosef served as vizier, the one who started the oppression ("a new king arose," Ex. 1:8) and who died (ibid. 2:23), and the one whom Moshe confronted.
However, we find opinions in the Gemara and Midrash (cited in Rashi to both of these verses) that take these expressions nonliterally: that the "new king" was actually the same old king but with new policies, and that his "death" means that he was stricken with tzaraas (leprosy or some other skin disease).
So conceivably, according to those opinions it may have indeed been the same king throughout the Jews' entire stay in Egypt. In Midrashic terms that's not unheard of. We find, for example, our Sages (Bereishis Rabbah 85:4) identifying the king of Tyre against whom Yechezkel prophesied (Ezekiel 28:1-19) with King Chiram, the friend of David and Shlomo (II Sam. 5:11; I Kings 5:15ff), five hundred years earlier - or even with Chirah the friend of Yehudah (Gen. 38:1), 1200 years before Yechezkel's time.
Some of the commentaries take this sort of thing in the literal sense - that our Sages are telling us that these people actually lived for many centuries, much like the generations of humanity from Adam to Noach who lived 900+ years. Others see such derashos as expressing the idea that, for example, the kings of Tyre over many centuries all shared the same salient personality characteristics (even though, on the face of it, the pious King Chiram seems to be very different than the boastful "prince of Tyre" whom Yechezkel condemns), and that similarly the various Egyptian Pharaohs all had the same mentality.