Ancalagon vs Gothmog
 

The most-powerful Dragon vs the most-powerful Balrog. Who takes this?

Ok firstly - and this is a different topic, but I'll still mention it - it's not clear that Ancalagon was the strongest of the dragons. A good case can be made that he was stronger than Glaurung - the only other competitor for that title - but then again Ancalagon is said to have been the strongest of the winged dragons. Whether this means that there were non-winged dragons who were stronger or whether Tolkien just happened to write this passage without thinking about such things is unknown.
What does indeed favour Ancalagon is his capability to fly and his hotter fire than Glaurungs, then again Glaurung had some tricks up his sleeve, being able to enchant people and bearing the title of father of all dragons.
So although I also rather think Ancalagon was stronger, it is never specifically stated. (or at least I am not aware of any such place)

Now to your question... it is much more simple. Gothmog is said to have been alongside Sauron Morgoth's most powerful and most important servant. Now again once could debate whether Sauron or Gothmog was stronger, but to your question this is irrelevant.

Comparing Ancalagon and Gothmog the winner is clearly Gothmog.

True. I guess I always thought Ancalagon was greater than Glaurung.

Anyhow, where is Gothmog called "most powerful and most important servent". I have heard Sauon called the greatest servent, which is not neccesarily the most powerful, but I have never heard this of Gothmog.

So is it not a physical battle then?

Whichever it is, Gothmog wins. He is a Maia and the leader of Morgoth's armies whereas Ancalagon was just a dragon, albeit a very powerful one.

Ares, I think you're wrong underestimating Gothmog's role.

Gothmog was High Captain of Angband and Marshall of Morgoth's troops making him basically the military second-in-command to Morgoth, whilst Sauron was also very important, but played more a domestic role rather than a front-line one.

So, these two together, were the most important servants.
Who was more important, is up for debate.

Single Balrogs can be defeated by determined foes (Ecthelion defeated Gothmog, and then there was Glorfindel and Gandalf). I think in single combat Ancalagon would beat Gothmog. After all, Earendil did not defeat Anacalagon singlehandedly, as he had the great eagles aiding him (I'm still trying to figure out how one stabs a winged dragon from the prow of a flying ship).

In any case, the point is moot because these two would never have come into conflict in a Tolkien-told tale.

Easy. He pairs with whoever the current Eagle leader is
and uses the Thach Weave :D . (see World War II
dogfights).

Meh... I wouldn't be so sure... I personally don't see how Ancalagon could defeat Gothmog.
Yes, Ecthelion did it, but he did so using a trick, noth through "normal" combat, I don't know how Ancalagon would manage that. I mean, his fire wouldn't be of use I guess, since Balrogs are creatures of flame and shadows. The only thing would be trying to bite him or maybe pick him up and let him fall from a certain height, but I don't see him being able to do either thing with Gothmog hitting him with his axe and the whip.

Well,since it is mentioned that Smaug could fly so quietly that he could avoid being noticed,why couldn't Ancalagon do so?Come ouy of the blue and grab Gothmog!

Anyway,althogh Ancalagon is more elegant,in any other term,Gothmog is superior!

Oh,and let us not forget the cases of Glaurung and Smaug-both killed by one single person,a Man.

What just came to mind whilst reading the last posts, is that the general vulnerability of dragons seems to play the decisive role in giving balrogs the upper hand in a battle.

Now, balrogs too are vulnerable, but usually it appears that it takes quite a lot to kill them. Never is a balrog killed with a sword, or an axe, or an arrow (at least not that I am aware of, and I am not taking early BoLT texts into account where they are much weaker). They always fall to their deaths - Gothmog, Glorfindel's balrog and the one in Moria all fall to their deaths.

Now dragons on the other hand don't need to fall from so high, you jsutn eed to find their weak spot - their underbelly. Glaurung died because of it, Smaug also, and I daresay that this is a very likely way of how Earendil defeated Ancalagon.

So the question is - how could Ancalagon catch Gothmog and drop him from high above without getting a big axe stuck in his underbelly? I personally don't see that happening easily, especially with the trusty whip with which Gothmog could hold on well.