Acquiring one of the really good spells for the first time. For example, ones of the spells that the wizard gains is Disintegrate, an arcane laser that cuts through swathes of enemies like soft butter (in normal difficulty at least, and that is when you will first unlock it). A couple of levels later, the wizard learns to use both hands so there are now two mystic death rays burning a path through everything in front of you.
While most of the awesome is reserved for the main cast in some fashion, the Mauve Shirts get their moments, too. By way of an example, consider Clyfton, one of the soldiers in Bastion's Keep who's part of one of the catapult crews during The Siege. In his own words:
Clyfton: "...Everyone else has gone and died, but I'll be damned if I leave my post!"
He then proceeds to work with you to raise the catapult, hurling vitriolic encouragement at you all the while. Consider that he's got the guts to do that while demons are breathing down both your necks, instead of panicking like most folks would do.
Malthael's introduction as the new villain in Reaper of Souls, he effortlessly takes out the Horadrim guarding the Black Soulstone and subdues Tyrael without taking a scratch. Then, he delivers this truly chilling Badass Boast:
Tyreal: The Nephalem...will stop you.
Malthael: No one can stop death.
During the final fight with Diablo, he loses patience and tries to get rid of you by sending to you to the Realm of Terror, a place where you can only get out by defeating his shadow. According to him, nobody has ever succeeded in doing that... and you still manage to do it. Cue Diablo entering a Villainous Breakdown for the first time.
The Templar Reckoning, the sidequest for Kormac, in contrast to the other two follower quests, is made of awesome. It has you and Kormac facing down the Grand Maester of Kormac's Templar Order and making him and his Inquisitors answer for the horrible things they have done to every single person they have made a Templar and stopping his plan to make Templars out of every citizen of Westmarch and beyond, with everything that this entails.
After inflicting a significant amount of damage to Malthael in Reaper of Souls, he will remark that the Player can not win - but his voice is full of doubt and uncertainty. Your character is such a badass that even the Angel of Death starts to realize that he's probably not going to win the battle.
Defeating Malthael in and of itself. The one who was regarded as the leader of Angels, and who even Imperius considers with a wide degree of respect. Add to it that he infuses himself with the power of the Prime Evil itself, the fact that you still manage to take him down is really quite incredible, a fact that Tyrael himself notes.
Freeing Auriel in Act IV restores hope not just to the angels, but also to the people in Bastion's Keep. For you, it's just a trip back to sell, salvage or stash all the loot you have gathered; for the people in Bastion's Keep, you're doing the impossible: taking the heavens back from Diablo, and not just making it back alive, but also winning.
And every step after that, Diablo taunting you with how hopeless it is, and how your efforts are useless, and you match it with proving him wrong each time, up to and including kicking his misappropriated ass.