Would have to put David Bronstein on the list. He came so close to world champion in 1953 Zurich. A strong theorist who was famous for his contributions in Kings Indian.
The top 10 most famous chess players that never became world champion

In my opinion,Bronstein,Keres,Stein,Korchnoi,Rubinstein,Schlechter,Bogoljubov,Reshevsky, Ivanchuk,Shirov.

I would like to add that the reason Morphy is not on my list is simply because nearly everyone considers him the unofficial world champion of his time.

Good picks Apotek. I really like that you included Stein, a seldom recognized player who almost made it to World champion.

Thank you!yes,the great Stein was clearly World champion material!And had he lived longer who knows what heights he might have attained..

I want to do something out of the ordinary and elect Tal and Spassky's trainer and second Koblents for his wonderful contributions to chess; and yes he was a master.

If FIDE champions don't count as world champions, then Topalov is definitely near the top of the list.

"Most famous" means you just have to pick well known players who didn't become champ. I will try to stick to a list of famous and pretty good.
Rubinstein, Tarrasch and Nimzo are probably the best picks here. I'm not sure how famous Stein is. He's up there with Geller in terms of guys who were close but not quite, but not particularly famous either, at least in the west.
Yusopov was close and topped out at #3 in the world I think. He's kind of well known these days.
Pillsbury is on the list of players who were strong enough to be champ, but it didn't happen because of an early death. Not super famous compared to some of the others listed. Sort of a predecessor to Stein.
Korchnoi, Keres, Bronstein, Najdorf and Larsen would be my others off the top of my head.

I think you should limit it to players who never had a chance to play for the title. Korchnoi had two chances, Bronstein and Tarrasch one.

Nimzo was exremely influential but as a player he was not world champion material.As for Stein,he was stronger than Geller .Najdorf and Larsen,sympathetic choices but hardly world champion material.Geller or Stein were way stronger.As for Tarrasch,he was Lasker's best client!

I'm picking only strongest non-World Champions. I don't care whether you've heard of them or not.
H.N. Pillsbury, A. Rubinstein, R. Charousek, J. Szen, E. Bogolyubov, V. Topalov, J. Polgar, S. Reshevsky, V. Korchnoi, D. Bronstein.
These are the first ten that come to my mind.

I'm picking only strongest non-World Champions. I don't care whether you've heard of them or not.
H.N. Pillsbury, A. Rubinstein, R. Charousek, J. Szen, E. Bogolyubov, V. Topalov, J. Polgar, S. Reshevsky, V. Korchnoi, D. Bronstein.
These are the first ten that come to my mind.
I'd probably take Judit Polgar off that list. I mean... she qualified for the Candidates' once, and didn't do so well. Guys like Grischuk and Gelfand have come closer to becoming World Champion.
Since the top 10 strongest chess players that never became world champion has been asked before even though not oftenly, i would like to ask a similar question.
Obviously, if i ask you guys the top 10 most famous chess players then most of the answers will be filled with World champions.
That is why I exempted World Champs from this list.
The chess player has to at least play like a master level of his time, not some celebrity patzer that happens to play chess.
I hope this thread will be interesting.
Before someone asks, i guess Paul Morphy could be on this list but w/e its your choice if he counts or not.
You can also mix players of different eras of course.