I'm wrapping up my level one save, and aside from the daedric quests, these are the quests that you can't get if you stay level one. I thought some people might like to know since the UESP and the wiki don't list them all properly.

• 1.) Dark Ancestor, from Dengeir of Stuhn.
  
• 2.) Kill the Vampire, from Sybille Stentor
  
• 3.) Kill the Giant, from the Jarl of Dawnstar. --Could prevent you from becoming Thane.
  
• 4.) Grimsever's Return, from Mjoll the Lioness.
  
• 5.) The Wolf Queen Awakened, from Courier.

Am I missing any? As I've said, I'm not counting the daedric quests because most people who are staying level one would be well aware of those. The Dawnstar thane quest was particularly unexpected. I intended to side with the imperials, so I might still be able to become thane as a war reward, but both jarls offer the quest and I'm not so certain.

Edit: I just found another as I wrote this, and it's a big one, too.

• 6.) Dungeon Delving, Jarl Igmund. --Could prevent you from becoming Thane.

If I can't go get his shield, I can't become Thane. Those who side with the Stormcloaks might be able to bypass it as a war reward. I'm not sure. If that would work, maybe Imperials could do the same by handing over the city in the peace treaty and then taking it back...

My last two SC playthroughs, I've waited until after the (originally) Imperial cities were taken before going there and starting any Thane stuff - because it seemed like a waste of time to become Thane for one Jarl and then have to go through the dialogue a second time when the city changes hands. (Except for Solitude, where there is no new Thane anyway, and I have problems with the Thane quests there if I wait until after the SCs win the war; and Whiterun, where Balgruuf is gonna make you his Thane whether you like it or not).

In Markarth, I found that there was absolutely no special quest that had to be done for the new Stormcloak Jarl in order to get through the process. As soon as I entered dialogue with him the first time, I was offered the chance to by the house there as a reward for my military service. Once I did that and went back to ask him if he needed anything else, I was told there was room for a new Thane, but I must own property in the city and do nice things for the local peeps, yada yada yada. Those requirements were already met and when the Jarl was informed of it he made me Thane.

I'm pretty sure the same thing happened in Falkreath and Morthal, although of course I didn't have to buy a house because there are none available in those holds. I know I didn't have to do the quest that Idgrod gives you if you want to get Thaned by her, and I certainly don't remember having to do anything special for the new Jarl in Falkreath either. I would approach and IIRC immediately get the "is there anything else you need" option and that would lead to the Thane stuff.

Whether this works the same way with Dawnstar, Winterhold, etc., if you play the Imperial side, I don't know or can't remember - I've done the Imperial side of the war but never became Thane in most of the holds after finishing it.