Not necessarily asking for your favorite moment, but the moment where you genuinely felt like the badass the dragonborn is supposed to be.
Mine was when I encountered my first flame atronach. I had read that they explode when killed, so just as I delivered the last blow I sheathed my weapon, went into third person, and slowly walked away from the blast.
My first wild dragon kill. This was my first character (obviously), a Paladin, and I was climbing the steps to High Hrothgar. The whole time I could hear a Dragon roaring in the distance, but couldn't see it. A few minutes up the mountain I looked over Ivarstead for the view, to see a dragon perched on the inn breathing fire across the street! Making a rapid descent down the mountain, I charged into town with my sword and shield, and helped the guards slay the beast. Even got that head-thrust killmove on it!
One of them is probably when I was on my dark elf archer char, and i took out 6 bandits rapidly, without missing, i was like :O, felt like a total legolas. oh and with my warrior character, using sweep on like 3 thalmor and killing them. Theres so many moments since iv played 100's of hours o.o.
When my lvl 7 Dunmer assassin killed a dragon with a giant on it's back. I killed the dragon while the giant paratroopered inside the Imperial fort near Whiterun. I killed the dragon and afterwards ran to the fortress as fast as I could to help the legionaires kill the giant. 2 brave men, a giant and a dragon died that day.
On my pure mage character, I was in a draugr ruin. I was surrounded by undead and low on health. I cast the circle of protection and they all crowded around me. using my last magicka potion I scrounged up enough magicka to cast the master level fire destruction spell. it charged up just as my circle fell and I unleashed the energy. every one was blasted back, incinerating all of them.
My favourite badass moment has to be when I was doing the dark brotherhood missions for the first time and had to kill the bride while she gave her speach. I snook up the stairs behind her and FUS RO DAH'd her off the balcony. Then I drew my sword and leapt after her laying waist to the guards. I killed about 6 guards decapitating the last one. then I put my sword away, cast invisible and walked out.