The Arithmancer/Awesome


 * Hermione making the second set of anti-Basilisk glasses by computing the spell to color them correctly in her head and composing an incantation in Dog Latin to match its required rhythm -- in the space of about 10 seconds. While the gobsmacked Arithmancy professor watches her do so.
 * As the last task of her Arithmancy OWL at the end of her third year, Hermione is told to modify the Tarantellegra ("dancing feet") spell to make a dummy perform a waltz. The examiners expected her to work out the changes on a handy blackboard before trying the modified spell.  Instead she computes the entire thing in her head, and finds an appropriate English phrase to use as the incantation ("Waltzing Matilda"), and joins the dummy in a brief dance, leaving the examiners astounded and congratulating her on doing something never seen before in any exam in their experience.
 * Hermione scores the highest score ever on both her OWL and NEWT Arithmancy exams -- the latter of which she takes at the end of her fourth year.
 * In sixth year, catching a hex from Pansy Parkinson on the tip of her wand, taking it apart with a second wand, putting it back together as a taser hex, and flinging it back. And then she follows it up with a massive show of special effects while reciting Galadriel's "dark queen" speech from The Lord of the Rings.
 * Hermione on a broom, launching herself into a high ballistic trajectory through falling stalactites and other cursed defenses to get to the island in the middle of Voldemort's cave. Later acclaimed in-universe as an awesome thing.
 * Taking down Bellatrix Lestrange with magical nanites that consume her flesh to create more of themselves.