I was thinking the other day, which were the 2 most powerful entities/characters that came into direct combat with each other. I know that the Gandalf/Durin's Bane battle may have been the instance of this in the Third Age, but what of the other Ages? Is the Durin's Bane/Gandalf truly the most powerful 1v1 battle to take place in the Third Age?
While it didn't end in a 1v1, Melkor vs Ungoliant did have a brief 1v1. Melkor was a Vala, and thus stronger than a Balrog who is only a fallen Maia, and at the point of their battle Melkor was afraid of how much power Ungoliant had, to the point of summoning his Balrogs with his screams. Neither of them won a victory, Melkor was wounded and Ungoliant fled, but there aren't many entities more powerful than those two in all of Arda.
As requested in the comments, here is the quote, however it doesn't mention fear specifically:
Then perforce Morgoth surrendered to her the gems that he bore with him, one by one and grudgingly; and she devoured them, and their beauty perished from the world. Huger and darker yet grew Ungoliant, but her lust was unsated. 'With one had thou givest,' she said; 'with thy left only. Open they right hand.'
In his right hand Morgoth held close the Silmarils, and though they were locked in a crystal casket, they had begun to burn him, and his hand was clenched in pain; but he would not open in. 'Nay!' he said. 'Thou has had thy due, For with my power that I put into thee thy work was accomplished. I need thee no ore. These things thou shalt not have, nor see. I name them unto myself for ever.'
But Ungoliant had grown great, and he less by the power that had gone out of him; and she rose against him, and her cloud closed about him, and she enmeshed him in a web of clinging thongs to strangle him. Then Morgoth sent for a terrible cry, that echoed in the mountains. Therefore that region was called Lammoth; for the echoes of the voice dwelt there ever after, so that any who cried aloud in that land awoke them, and all the waste between the hills and the sea was filled with clamour as of voices in anguish. The cry of Morgoth in that hour was the greatest and most dreadful that was ever heard in the northern world; the mountains shook, and the earth trembled, and rocks were riven asunder.
The battle between the 2 most powerful entities was before Melkor's first capture. After hiding in the deepest pits of Utumno, Tulkas went and wrestled with him and threw him down.
But at the last the gates of Utumno were broken and the halls unroofed, and Melkor took refuge in the uttermost pit. Then Tulkas stood forth as champion of the Valar and wrestled with him, and cast him upon his face; and he was bound with the chain Angainor that Aulë had wrought, and led captive; and the world had peace for a long age.
The Silmarillion: Chapter 3, Of the Coming of the Elves and the Captivity of Melkor
Tulkas was the greatest fighter in the history of Arda, feared by Melkor himself, and Melkor being the greatest of all the Valar. This certainly was a fight between the 2 most powerful entities in the history of Arda.
I believe that Tolkien several times calls Melkor/Morgoth the most powerful being created by Eru, the most powerful of the Ainur, the most powerful of the Valar, etc.
Therefore the most powerful possible pair of combatants would be Eru and Melkor/ Morgoth.
So I suggest that the most powerful battle would be:
1) Melkor/Morgoth vs Eru, if the whole of the Elder Days is considered to be Morgoth's rebellion and figurative "battle" against Eru.
2) Morgoth vs Tulkas, as in Edlothiad's answer, if it has to be a literal battle and if Tulkas is more powerful than Ungoliant.
3) Morgoth vs Ungoliant, as in Daishozen's answer, if it has to be a literal battle and if Ungoliant is more powerful than Tulkas.
In The Book of Lost Tales - IV The Chaining of Melko - a battle is related between the Valar and Melkor/Morgoth featuring direct combat between Manwë and Morgoth.
"Then Melko sprang to his feet shouting in a loud voice and his folk came though all those dismal passages to his aid. Then lashed he at Manwë with a flail he bore, but Manwë breathed gently upon it and its iron tassels were blown backward..."
Manwë and Melkor are canonically the two most powerful beings in Arda so this would have been the highest powered direct combat.