Definitely Harry will never be as great a wizard as Dumbledore. That is more or less Hermione's role in the future all though I don't think she could ever fight like him. Harry while never willing to give up does not have the drive to study like Dumbledore or even Voldemort. At best he could fight at their level but he could never be as well studied as them. I always thought he was about the top tire of average like Ron.

 

Hermione isn't on Dumbledore's level either - more like on McGonagall's.

 

I've never quite agreed with the assessment that Hermione is bad at duelling. Certainly she's not as good as Harry, but I think by the time DH rolls around she's certainly better than 99% of the population.

Yes, in PS she panicked about not having matches, and in PoA she lost it with the boggart... but that was a long time ago. A time before the DA, duelling in the Department of Mysteries, duelling in the HBP Death Eater invasion of Hogwarts, duelling in the Battle of Hogwarts. She's got a lot of experience now.

So yeah, I think Hermione = McGonagall in terms of skill and duelling ability is a good comparison, though McGonagall of course has more experience.

Regarding Harry, I think you can definitely make a case by the end of DH that he's pretty badass, and could probably beat someone like Bellatrix or even Snape in a fair duel.

This requires a generous interpretation of the events of DH, but it is a viable interpretation. It's not completely against the books.

- Firstly, you have the point I made in the other thread (Just how good was Harry?) about Harry having passed Charms, Transfiguration, Potions and DADA was moderately good marks, especially in DADA. That means that he is able to do almost all magic, and though some of that magic may be imperfect, practice will smooth out those imperfections, as will the superior understanding that comes with age.

By the end of DH Harry is performing magic he would have strugged with the year before, even though he hasn't had any formal schooling in the interim. Which brings us to number 2.

- Secondly, by the end of DH Harry seems to have completely mastered non-verbal magic, which is one of the things that makes him look rather average in HBP.

- Thirdly, it could be argued that Harry has mastered occlumency and legilimency by the end of DH, capable of blocking Voldemort from his mind and diving into Voldemort's mind at will. This is a substantial skill. (I realise I have argued against this in the past, but we're going with a generous interpretation of canon here).

- Fourthly, Harry has always been very good at DADA and duelling especially, which speaks well of his skill as DADA involves a lot of Charms and some Transfiguration. Going by the "magical person" theory of magic I recently posted about, I think Harry is a highly magical person thanks to his willpower. Just look at the shield he casts just before his final showdown with Voldemort, covering the entire Great Hall.

- Fifthly, dying and coming back seems to have gifted Harry was a kind of enlightenment about fundamentals of magic, as suddenly at the end of DH he's able to understand events much better than Voldemort.

 

Agreed, the Hermione and McGonagall comparison is probably the best supported by canon.

Of course you're correct, that you could interpret Harry's skill level that way, but in my opinion this borders on wishful thinking.