Some ideas from a similar thread and Skyrim DLC and so as not to repeat some of the great ideas already stated -
Quests! Long quests, short quests as long as they're good quests (I am happy for them to involve new or existing factions and take place in Skyrim or a new/previously seen province) - choices in how to approach quests and complete them is always good.
Add more vampire specific quests (and maybe some Vampire specific enemies).
Maybe more Thalmor interactions (against them and for them - give my Thalmor sympathizer/secret agent character something to do - I believe the next game will probably be Thalmor centric, but till then it would be nice to do something more with them).
Crazy maybe - another transformation or two (Lich and Werebear perhaps?)
And some more meaningful interactions with giants other than avoid/kill. They're intelligent, do they have dreams? Quests they would like done? I'll do your quests giants. Do you want to forge a Giant nation for yourselves? I'm in - sounds exciting, sapient rights for all!
More things to do with your home (agreeing with other peoples suggestions here) more displays, customization, easier placement of items etc. And contributing to the community of a city the house is in - take the orphanage and its four orphans. My good but eccentric philanthropist character has too much money and is looking for good causes to become a patron of. Improve the orphanage (think of all those immortal children whose parents died fighting dragons!) or become a patron for a theater for example.
Maybe a bit more body size/height/shape variation/customization (mainly for the beast races and because it might be fun to play as a Tyrion Lannister-esque character

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As the topic describes, let us as the TES community give suggestions and ideas to make the TES series even better. Post your idea's and suggestions and dont forget the small details!
I'll start it off with my first personal input.
1) More NPC's. Skyrim is empty in general. Add more people into the world to make it more lifelike. Think of trade caravans, adventures, travelers etc. These dont have to be quest people. Also more different voice acting! All this will add significly to the immersion of the world
2) More wildlife. I am missing wildlife! You see the occasional wolf, rabbit and deer but this is hardly enough.
3) Better dynamic world. The world should react to what happens in the world, espec the dialoque in skyrim lacks this feature. This will have a huge impact on the immersion and overall 'do what you want' feature of the world.
Please people refrain from any flaming or trolling in this post, let this be a long list that hopefully bethesda will have a look at.
Yar to all these, though I am kind of ok with wildlife as it is, for a frozen land like this. Maybe more deer (ok, I'll be honest, I don't know a huge deal about wildlife in Scandinavian or east European countries or what one might expect to see in numbers).
I've had the same follower for more than 50 hours of the game, I'd like them to acknowledge the massive adventures we've been on and maybe even start to like me more.
Indeed, more like a TES flavored Bioware companion.