The Best Player Never to Become World Champion?

Greco

I think akiba Rubenstein he was very good at the end game

Fabiano Fabiano Caruana

David Bronstein.

Edward Lakser was good, but not that good.

Play what a coincidence Batgirl you have Sammy rachesky pictus the one and so do I because he beat every world champion before he was champion while he was champion and after he no longer held the title no other chess player did this I had the pleasure speaking with him he finished second all the time and is under zonals and all that because he didn't want to become world champion he like his independence he analyzed the game for me that's how I spoke to him via the phone he lived in New York have spoken to you in awhile nice to see you online hear you remember me that guy that plays all the crazy openings and I played on chess master take care have a good day everybody

I see what you mean. He was up a piece and made the following move on 32 without sealing it (he had this option). This win (was lost in actual game) would have tied the match at 9-9 and sent it into overtime. https://www.chessgames.com/perl/chessgame?gid=1036366 

Chigorin was a hero to many Soviet GMs including Smyslov. He deserves to be better remembered. But I still don’t rate him as highly as Rubinstein, Bronstein, Keres, Korchnoi, or especially Ivanchuk.

There is a method of estimating previous chess players (before rating system) based on players who play ranked players, then working backward. Steinitz is estimated at 2826. Based on Chigorin's second match with him, and that he also outperformed him in some tournaments, he may also be around the same ranking.

https://en.wikipedia.org/wiki/Comparison_of_top_chess_players_throughout_history

Chigorin did very good in open games against Steinitz.

Obviously.  The former is 2750 performance.  The latter is 3000+ performance, depending how close to 100% you meant.

This would be my list of "best players that never became World Champion".

"Carl Schlecter. He was soooooooooooooooooooooooooooooooo close to beating Emanuel Lasker in their WC match"

It's still just one short match, just like Karjakin against Carlsen. Schlechter's (or Karjakin's) other results were never anywhere near those of for example Korchnoi, who was clear #2 in the world for a decade, and qualified for title matches by beating lots of top players in a bunch of Candidates matches.

Perhaps you should examine Schlechter's career more closely. In match play he drew Janowski in 1896, defeated Janowski handily in 1902, drew Tarrasch in 1911 and lost to Rubinstein narrowly in 1918 when he was about half dead. He won Stockholm 1906 going undefeated, he won Ostend 1906, lost only one game at San Sebastian 1911, plus many victories in lesser tournaments.

He has lifetime plus scores against Alekhine, Burn, Nimzovich,  Reti,  Spielmann, Steinitz, Vidmar and Tchigorin among others.

He was called the Drawing Master, but he was actually more of the Rarely Loses Master, like Petrosian.

”Perhaps you should examine Schlechter's career more closely”

”He won Stockholm 1906 going undefeated”

"lost only one game at San Sebastian 1911"

I still don’t rank him anywhere near Korchnoi. If you bring up Stockholm 1906 on his list of achievements on the subject greatest ever non-World Champion it’s also worth mentioning the level of opposition :-) He shared first with Bernstein, and the other participants were far from elite players. San Sebastian 1911 was no bad result, but Rubinstein was undefeated and finished well ahead of Schlechter. Nothing wrong with the latter, but he was no Korchnoi or Keres. I’d place him a bit below for example Aronian too, but that doesn’t mean that he was anything else than a great player. 

PS. Edited to add some info on San Sebastian 1911

"the point is that Lasker completely dominated the matches against other players, but here he was nearly defeated by Schlecter, having a -1 score at one point during the match. This shows you how good Schlecter was compared to other players of the time"

I'm still no big fan of picking one result to assess a player. Schlechter played a great match against Lasker, and also had results like this around the same time:

Nothing wrong with Schlechter, but his being ranked #6 by Chessmetrics at the time of the Lasker match is reasonable. He was a great player but for example Rubinstein was consistently stronger back then.

Smyslov among others was a fan of Schlechter. Schlechter’s réputation as a draw master was a bit unfair. And he contributed quite a bit to opening theory. His Schlechter Slav has all but refuted one of White’s move orders, giving black instant equality, and is still used today in top flight chess.

I believe that Schlechter was hard done by in the match with Lasker, but I also believe that Lasker preferred to play Schlechter over Rubinstein. Rubinstein was definitely better than Schlechter.

As I said earlier though, Schlechter deserves to be remembered better than he is. He played quite a few memorable games.

White achieves 68.5% against the Schlechter Slav in the chess.com master database. That means if it is a route to equality, it is a route that an awful lot of strong players don't find.