1: Courier messages. Your character in Skyrim will get personal messages from a courier who runs up to give it to you. This is usually something random, but still makes the world feel alive and like your character is important, and that he/she is being watched by the powers that be.
This happened in Daggerfall quite often, too. I'd be in some random city, when I get a message saying that a shadowy figure, or perhaps a small child, runs up to me and hands me a letter. Occasionally it was even a carrier bird, and one time even a zombie with a letter pinned to it.
2: Assassins/thugs. Sometimes in Skyrim, depending on what you've done or who you've sided with, you might randomly encounter assassins or thugs that were sent after you from certain people. This happened in Daggerfall a lot too, where the assassins might cry out something like "You never should have crossed [insert faction or NPC name here], [player name]. Now you pay the price" or something like that.
3: Randomly generated quests that never stop coming. In Skyrim with the new Radiant Story system, you'll never stop finding bounties to fill, or miscellaneous favors to do for people, etc. Daggerfall did this too (in fact all the quests aside from the main quest were always randomly generated, which did get old after you saw the same quest 10 times).
The reason I am so pleasantly surprised by all this is because what Daggerfall really captured for me was the feeling of a living, breathing world. Stuff changed even when you didn't have anything to do with it, people did what they did without you telling them to, and most of all, factions and important people recognized you for your power, allegiances and fame (or infamy) and responded to that.
Now, Skyrim isn't perfect, but by recapturing this living, breathing world feel that in my opinion only Daggerfall previously achieved is something truly awesome.
Anyone else feel similarly?