What is the most difficult winnable endgame to learn?

Title says it all. What is widely considered to be THE most difficult WINNABLE endgame to learn?  B+N vs. K?  N+N vs K+P?  Something else?

I'd rank them like this, easiest to hardest:

B+N vs K (can be really confusing at first, but after regular practice can become very easy).
N+N vs K+P
B+R vs R
Q vs R
B+B vs N (this comes to mind as the hardest "basic" winning endgame, don't know what others may pick. By the way this was thought to be a draw until computers helped).

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For ones with pawns, various R vs minor piece false fortresses come to mind as difficult. Also the Q vs R+P false fortress is difficult to break down correctly. Although for these, it's also hard for the defender, so I wouldn't call them harder than the pawnless ones. For the pawnless ones, the defender only needs to know a few basic ideas to make it difficult.

Oh, and for B+B vs N I'm thinking of the position where the knight can get to b2,b7,g7, or g2. IIRC this was thought to be a fortress until computers solved it as a win.

Interesting question, since you're asking for endings that are hardest to learn, rather than hardest to play once you've learned them. For the pawnless endings, I agree with putting B+B vs. N at the top; I'm not sure that ending is even possible to learn. But for me, R+B vs. R seems at least as easy to learn as the basic mate with B+N. The drawing procedures (Cochrane defense and second-rank defense) aren't hard to learn. In the rare cases where the stronger side can win by reaching the Philidor position, the win is fairly lengthy but quite logical and not too difficult to learn. I think N+N vs. K+P is quite a bit harder to learn, due to all the cases that have to be studied when the pawn is about to cross the Troitsky line, not to mention the exceptions where the knights can win even when the pawn is well beyond the Troitsky line.

That's another one that probably can't be learned. But depending on how you like to spend your time, it can be fun to play around with using the Nalimov tables, to see if you can find a position that gives the greatest distance to mate (238 moves).

I believe studying rook endings is more beneficial than studying any other endings . Why ?  Because of their difficulty and their frequency . I have read that about 50% of all endings are rook endings . 

B+B vs N is possible to learn, I've known people who have spent a long time learning it... I don't think it's a good way to spend your time unless you're a professional.

I didn't know R+B vs two knights was a thing (I don't have e.g. secrets of pawnless endgames).

Q+N vs R+B+N is winnable

Of all the basic mates, K+Q vs K+R seems to be the most difficult. Even World Champion Candidates get this one wrong. 

Slightly more complex endings such as K+R+B vs K+R are also difficult, but if we limit basic mates to those with four pieces or less, it's K+Q vs K+R.

I agree completely! Well, K+P endgames are equally essential, and are important for developing other skills such as visualization and calculating long lines.

But of all the "piece" endgames, Rook endings give you the most bang for the buck.

K+R+b vs K+r+b

Here's a game between Timman and Speelman that may or may not be winnable. The problem is the 50 move rule. With best play from both sides, there are positions that are winnable, but that take more than 50 moves to prove. 

n+b mate

1. Knight and Bishop Checkmate

2. Double Bishop Checkmate

3. King and Two Pawns Vs. King and Pawn

il say q vs b+b

I had a B N v King the other day. I couldn't close the game. Afterwards I studied it. It is difficult to do but the hardest part is how seldom it happens. It's hard to remember when it only happens every few hundred games. If it happens tomorrow I will win but next year???? will I remember??

King + 2 Knights vs King and Pawn. Here's an example forced mate in such an endgame:

Quite hard are the endgames of the type R + 2 light pieces versus R + 1 light piece. They are long wins but they commonly do win.

If you ignore the 50 move rule then there are a number of very long tablebase wins, some of them would take well over 500 moves if you played perfectly and the opponent also defended perfectly.

I wish the 50 move rule was at least 100 moves. That would be more reasonable. The longest pawnless Endgame forced mates take up to 550 moves.