I'm almost hesitant to bring this up since I recall it only vaguely from high school -- I don't know what status it has theologically or who said it: that Christ did not have to die on the Cross, a criminal's death, utterly painful and humiliating, to save us. By this "theory" (if that's what it is), the Incarnation would have been enough. But the overflowing generosity and love of God doesn't settle for "enough" -- it goes for abundance!
I do know that when I've been going through the greatest physical pain, I've also felt closest to God -- and this holds even for far lesser things like self-denial during Lent.
Amazingly, we have a sort of religious debate here without any rancor, and you and xzins have both made excellent points. Fasting does raise the question on whether self-induced suffering is something to be sought after. But fasting, as you mentioned, is a type of self-denial rather than an active means to induce pain. I am not able to recollect any instance in the scriptures where the righteous actually inflict physical pain upon themselves, but many instances where, for a righteous cause, they endure pain inflicted by others. Jesus certainly allowed the nails to be driven into his hands, as he could have easily called upon a legion of angels at any moment to overcome his tormentors, but I don’t see this as self-inflicted suffering even though it was self-permitted. Also, the suffering on the cross had an underlying purpose, as it was a means to bring salvation to a world of sinners. In a similar, if infinitely lesser example, someone undergoing a painful recovery after an operation to donate a kidney to a loved one is enduring a self-permitted type of suffering, but the suffering is for an underlying purpose, and is not done simply for the sake of the suffering.