My theory based on the first episode is that Doom will slowly fall for the cutie that is Tak Dong Kyung, he keeps pestering her to make  a wish and after being around him for a while and seeing how hurt he is in his eternal life, she wishes for him to be free of being Doom and dies hoping that when Doom dies a natural human death they can be reunited in the afterlife.

I have another version of the above theory that after her wish, Dong Kung goes into a coma and Doom desperate to help her seeks the help of the girl in the hospital and after making a deal, Dong Kung makes a miracles recovery, several years later they are reunited again in a heartfelt scene after Doom has finished the deal he made, and they get their happily ever after.

Him saying that he will give the doom/death to whoever she loves most is making me go, "WhELp see ya later Mr Doom when we get to episode 16, I'll get my tissues."

All of these suggestions are so good - spot on! Personally I think the most likely one might be that they meet again in another life. But I would like the suggestion of: "She dies but becomes doom along side him and they doom people together for eternity". I really like that idea too! It is also a bit refreshing since I can't really think of any dramas where the human lead "joins" the non-human one in their lifestyle. It is always either both living as humans or one living on as human while the other remains non-human. (Sidenote, if anyone knows of a movie/drama where the human becomes non-human for their partner please tell me :)

My take on the ending: I feel like they are trying to be a bit innovative and non cliché. Primarily the mood of the drama makes me feel like they are trying  to create a story which is refreshing and bold. So I think no matter the ending it won't be too predictable and perhaps it will be sad. 

Wouldn't it be a bold move if her wish managed to free him but at the cost of her becoming doom instead? And the ending is her walking in the world as the new doom? 

This type of story/genre can lead to a handful of predictable endings, however, with the way the first two episodes went, I have a feeling it won't be as predictable as we might expect though I love the ideas tossed! Perhaps it's too soon to say that, but what I can say is that it may be bittersweet yet satisfying ending, kind of like Hotel del Luna or Goblin. But I must admit, I just love the idea of either both of them being "Doom" or, as @noyba mentioned, TDK wished Myeol Mang free and became the new Doom as a price to be paid. If that ever did happen, what a plot twist of the century it'd be, and we'd legit need a second season for that.

oh OH! "Then someone else will die instead of you. The person you love the most at that moment."

Okay so I think we know where this is headed then?  I mean assuming he's contractually obligated to destroy the world OR kill the person she loves and can't just stop it from happening. She falls in love with him and they are faced with destroying the world or killing him? I mean if he can just nullify the contract then my theory goes out the window. So he dies and we have some Goblin or Angel's Last Mission situation? 

 Dani:

"Wouldn't it be a bold move if her wish managed to free him but at the cost of her becoming doom instead? And the ending is her walking in the world as the new doom?  "

Oooh I kind of like the idea of a swap, the price of freeing him being taking his place. 

I know right! I absolutely don't think it is possible for that to be the ending but I kind of wish it were as that would be a very novel ending.

Unfortunately I think the hint of "her most beloved" dying is the answer... He will die instead of her and then either she will live a happy life without him or he is reborn as a human and they get together. I hope we are wrong though as that feels so predictable. 

Oh I love happy endings even if they are predictable! Especially since it looks like Youth of May could be tragic I can only take so much sadness!