Paul Keres! Michail Chigorin...
David Bronstein!
Best Player- Never World Champ.
Sort:
Paul Morphy! (They didn't have an official world championship back then)
Carlsen, probably. :)
Keres won more tournaments and matches, than anyone above the list...
Keres
was politics involved that these players didnt become #1 or was it that they just were not at their peak when they did have their chance?
Korchnoi and Keres
"Greatness" and playing strength are not necessarily the same thing. Nimzowitsch had an impact on the game that few others did. Korchnoi is amazing for his longevity (a la Lasker). He was a World Champion Candidate at age 60, and he's still playing competitive chess at a very high level.
Duffer1965 wrote: "Greatness" and playing strength are not necessarily the same thing. Nimzowitsch had an impact on the game that few others did. Korchnoi is amazing for his longevity (a la Lasker). He was a World Champion Candidate at age 60, and he's still playing competitive chess at a very high level.
While I agree with this duffer, it seems to me you are selling short Nimzowitsch where "playing strength" is concerned. At his peak I think he was in the top 5 in the world. He was certainly top 10.
I agree with those who posted Keres and Korchnoi. Great players!!
Anand.
ericmittens wrote: Anand.
Anand IS the world champion eric......pay attention ! 
ericmittens wrote: Anand.
I think Anand is a candidate for the distinction of player who would have been the strongest never to be world champion if he'd never been world champion.
I would agree with a lot of people saying Keres and Korhnoi.
Who do you think is the best chess grand master never to be world champion?