So far mine had to be my trip to Eldergleam Sanctuary. Just before arriving, I saw a dragon circling. At that point, I had only killed 2 dragons - both with the help of Whiterun guards. I didn't think much of it because it was my third dragon sighting on the trip and the previous 2 had been of the non-encounter, endless circling but never land variety. I was on fire before I even realized this one was going to be different.
I recovered from my initial shock, healed a bit and started attacking. Lydia and some guy who had asked to come along attacked it as well. (At least I think he attacked it although I can't say for sure what he did.) Anyway, the fight kept alternating between it landing and fighting, and sometimes taking off and breathing fire down. I would switch to healing whenever it took flight which seemed more productive than firing arrows. At one point I had just started to heal after it took to the air when I was suddenly on fire again. I could see the dragon clearly in front of me though and it hadn't recently breathed fire - that's when I turned around and saw the second dragon behind me. I was actually fighting two of them which is probably why the pace of the fight had seemed so hectic!
The entire fight took at least 5 minutes, maybe more, but at the end of it, I was standing between 2 dragon corpses a few feet apart. I turned to congratulate Lydia on a job well done, but she was nowhere to be found. Neither was the gentleman whose name I can't recall. I took a few steps to a rock that provided a higher vantage point to see where they might be and I located them both - lying dead behind that rock. I briefly considered reloading to my last save before the fight to return my Housecarl to life, but ultimately decided against it. She had died a "good death" in an epic fight, likely saving my life. I went home to Whiterun and closed the door to her room in our house and haven't opened it since.