One of the difficult tests Avraham Avinu faced was when Nimrod threw him into the (Kivshan Haish) fire. It is not mentioned in the Torah at all. Why?
Though Rashi (Avot 5:3) lists the Kivshan Haish among Avraham Avinu's 10 trials, the Rambam counts only the trials listed in the Torah among the 10 trials that Avraham faced. So in his view, Kivshan Haish was not 10 of Avraham's 10 trials.
Ramban (to Gen. 12:2) says that the Torah didn't want to go into everything that happened between Avraham and the idolators of his native country, because it doesn't want to have to explain their theological ideas.
In one of his talks (Likkutei Sichos, vol. 25, pp. 47ff), the Lubavitcher Rebbe zt"l explains that the Torah indeed wants to make it look as though Avraham was chosen by G-d for no discernible reason. This is meant to teach us that the chosenness and uniqueness of the Jewish People - of whom Avraham was the first - is ultimately based not on intellectual understanding of G-dliness (which indeed was Avraham's gigantic and independent accomplishment from age three onward), but because it is His choice and His initiative. Accordingly, our fundamental relationship with Hashem has to be based on simple and unquestioning kabbalas ol (acceptance of His commandments) - as demonstrated by Avraham's leaving his family and nativity, and venturing into the unknown at G-d's command. (In turn, he says, this also means that when you meet a Jew who doesn't yet know what it means to have a relationship with G-d, start with a practical mitzvah - even if he or she doesn't understand what it's about - and the rest will follow.)
We don't learn to our own lives things from the Ur Kasdim miracle. Since there Avraham started disputing (he broke the idols of his father), but for the halocha until somebody interfering our lives we don't have to proof to all that we are right. Avraham made so because it was his way to spread the monotheism in the world.
It is described in the Book of Daniel about the miracle of Hannan'ya Mishael and Azar'ya in many details, because that story learn us how to behave in situation when somebody do interfering our lives.
Same answer for many miracles that are not mentioned by Torah, like this one.
I think there are two important points to be made in considering this question.
1) Many of the insights provided to us by the midrashim regarding Avraham's "youth" (i.e. up to 70 years old, when we are told the first narrative information about his life other than birth, marriage and relatives dying) bridge the gap between the world of sin/non-recognition of God that we've known about since [Enosh or] No'ach and the rapid development of monotheism. This development, which was catalyzed by many possible epiphanic events in those 70 years, is not central to the Torah's primary message of carrying out service of God once He has presented us with requirements, or to Sefer B'reshis's primary purpose of establishing the Divine process of determining and delivering [contingent] stewardship of the Land by B'nei Yisra'el (for lack of a better term). The Torah almost never spends time on "character development" - expect it as one might from a literary narrative.
Perhaps to suggest a different approach. Perhaps the reason it is not mentioned in the Torah is because it didn't actually happen. It is a Medrash that demands explanation, but we are not bound to take it literally (see other responses on taking Medrashim litarally). Perhaps the interpretation can better be understood by understanding what Avraham did. Avrham grew up in a society of Pagan Idolaters. He through careful analysis and thought came to the conclusion that this belief system is wrong. He showed clear proofs and arguments as to the fallacy of worshiping idols. He was against the prevalent religious view of the times. He began to share his views with people. He began to have a following of people (Rambam reports thousands). This put him in a very dangerous position. Nimrod himself according to the commentary (see seforno, Bereshit 10,10 on Tower of Babel) used the peoples religious beliefs to gain power over the people. Avraham was a threat to his plans to gain power. Perhaps the Medrash is suggesting that it was due to Hashem's assistance that Avraham was able to navigate the political terrain and survive. Exactly what the situation was, I don't know, but it was a dangerous political situation and one wrong move could have been the end of Avraham. We have seen many philosophers confronted with similar situations in history. So, it doesn't mean literally a oven of fire, but that Hashem assisted Avraham from the political fires that threatened Avraham's life. (See Rambam Hilchot Avodat Chochavim Chapter 1, Law 1)
I heard a nice (cute) Pshat on this. By all other Nisyonos, or tests, the Satan worked hard to dissuade Avraham from coming through. By the fire, the Satan couldn't have been happier than to see Avraham burn and all his life's work go up in flames.
Personally though, I don't like this reasoning because the Satan has a job to do. He was created to be a test and an obstacle for people to overcome, not to work against Hashem.
I would suggest another reason for it not being included. The threat of Nimrod to toss him into the fire is not a test of the strength of his faith, but rather a test of what he is willing to endure for the sake of his faith.
All other tests were not necessarily hardships per se as much as a test to his faith. In the case of his showdown with Nimrod he was standing up for his beliefs and ready to die for it. Idealistically this is easier than having followed Hashem's personal command to settle in Eretz Yisroel only to have to leave there to go down to Egypt. This is a test to his faith in Hashem to see if he would question Hashem. Lack of a miracle when he got himself into trouble is not a test to faith.