Adolf Anderssen was the best player in the world for a lot of his life, but was never 'World Champion' because there was no official match or organisation to arrange an official match.
He was a wonderful attacking player, but lost matches to Morphy (1859) and Steinitz (1866), who both knew how to attack, but understood more than Anderssen about developing and defending. But he was a great favourite, not just among chess fans, but also chess players, for being a decent chap as well as a terrific player.
Frederick Edge describes Anderssen:
Xavery (Saveilly) Tartakower was the wittiest of masters, and, it was said, "too in love with chess to ever become world champion", as he would often play an interesting move over a plain one, and perhaps throw the result into question. He was the champion of chess journalists, and his epigrams, or Tartakowerisms, will be quoted as long as chess is played. Everyone loved to talk with Tartakower, not just for his wit, but also his great knowledge and appreciation of books and art.
"The blunders are all there on the board, waiting to be made."
I don't entirely like the parade of endless men in the 'Lessons from...' series, so here's one from the other half of humanity, and a fine Appendix to the list of guys that ever joined the 'Vera Menchik Club'.
Judit Polgar was strongest woman chess player ever. She never became World Champion, and was never interested in becoming World Women’s World Chess Champion ( a title held by her two sisters, Susan and Sofia). She was in the world top ten and improving when she retired.
The first Hastings tournament in 1895 was the strongest tournament up to that time. Every top player was invited and they all turned up: World Champion Lasker, ex-champion Steinitz, Steinitz' challengers Chigorin and Gunsberg, Lasker's future challengers Schlechter and Janowsky, the old British Champions Bird, Blackburne and Burn, and the new US Champion, Pillsbury, whom no-one knew much about, but he has been taking lessons from Steinitz...