I find myself using Azura's Star every time it is full to level up enchanting. Currently, I am using the arcane enchanter in Dragonsreach which is quite a walk. What is the closest arcane enchanter to a fast travel drop point? Are there any in the open world so I don't need to enter/exit the area every time?
I like to use the arcane enchanter in Drelas' Cottage. Its easy to fast travel to and has both an enchanter and alchemy station. The only draw back is that you do need to cross a loading boundary and that the owner is a rather high level mage that does not take kindly to strangers.
If you are playing Skyrim on steam, there is a nice mod in the workshop that adds a fast travel icon on the world map for every player owned house.
The fastest one to travel to that I know of is in Mistveil Keep, in Riften. There is an arcane enchanter just inside, in the first room on the left.
The house is just a short walk from the entrance to Riften, and the Enchanter is just downstairs. If you're Thane of the Rift, you've also got your handy-dandy Housecarl, Iona, available if you need help carrying stuff. And, if you're also inclined to do some Alchemy, you can buy another upgrade for the house that puts an Alchemy Lab right in the next room.
Like I said, it may not be the absolute fastest but, I find it fairly convenient. It's the one I personally prefer to use.
NOTE: The link for "Enchanter" above has a list of all the Arcane Enchanters in Skyrim.
Found one really close to fast travel point, no fight and no loading screen.
It's a place called Sacellum of Boethiah (got there following quest lead actually) here it is on world map: (click for full image)
Here is the enchanter in a tent up some stairs: (click for full image)
And here is the enchanter location in local map: (click for full image)
I find Orphan Rock just southeast of Riverwood is the fastest Arcane Enchanter. It's out in the open, and there's no loading required to reach it or tough characters surrounding it.
In Windhelm. There is an Arcane Enchanter outside behind Niranye near the Blacksmith forge on the west side of the city.