Well... if we define the artist as "successful" who not only does incredibly well in financial terms but also has attained such a level of critical esteem that he or she is almost certainly assured a place in the narrative of art history then the "most successful" would almost certainly include:
Johns is in the $5 Million US per new canvas range and has a waiting list. Older paintings by Johns have sold for over $40 M. He was a central figure in the shift from Abstract Expressionism to Pop Art. His works include some of the most iconic pieces by any living artist and his works can be found in nearly any major museum collection.
Lucian Freud is also into the $5 Million plus range for any new canvas. His 1995 painting, Benefits Supervisor Sleeping
recently sold for over $33 Million. He is seen as one of the last living "old masters" and his paintings have have a profound influence on subsequent painters including Jenny Saville, Anne Gale, Cicily Brown, Alex Kanevsky, etc... Like Johns, his work is avidly sought by museums around the world.
Kiefer is perhaps the most towering figure to come out of the 1980s art movement known as Neo-Expressionism, and the most important Post-War German artist. He almost single-handedly shattered the premature declarations of the "death of painting" and brought a profound seriousness and ambition to painting once again with his vast tortured canvases dealing with the traumas of history... especially the Holocaust. Kiefer's paintings sell in the $2-4 Million range and are actively sought in major museums around the world.
Brice Marden and Cy Twombly:
Twombly and Marden are both in the top tier in terms of price per new canvas. This is owed much to their perception as the last living links to the heroic generation of the Abstract Expressionists. Stylistically, they both are deeply indebted to Jackson Pollack. They are both highly sought by collectors, but do not seem to have as firm a hold on a position in art history.
If one were to add a third to Twombly and Marden it might be Sean Scully:
Scully is seen as having successfully merged the polar opposites of Minimalism and Abstract Expressionism... bringing a warmth or passion to the cold logic of Minimalism and a rational structure to gestural field painting. Along with Marden, Freud, and Johns he is one of the few living artists to have enjoyed a retrospective at MoMA or the Met while still alive.
Gerhard Richter would be another among the most successful artists:
Richter brought a certain European detachment to painting... developing several very different signature styles ranging from conceptual art to "squeegie-made" abstractions to photo-derived realism employing blurred photo references. Like Scully he has also been awarded the major museum retrospective. His paintings are in the $5 Million range... and his influence on a whole generation of subsequent photo-realists is undeniable.
Finally, I would include Chuck Close:
Close sells within the $2-4 Million range. His stature zoomed after he heroically continued to paint following his spinal artery collapse in 1988 which left him severely paralyzed. At that time his work took on an increasingly colorful and "expressionistic" edge, although it maintained its link to photo-realism and "process painting". Close's selection of well-known subjects including Lynda Benglis, Cecily Brown, Paul Cadmus, Francesco Clemente, Bill and Hillary Clinton, Merce Cunningham, Willem Dafoe, Eric Fischl, Philip Glass, Al Gore, Jasper Johns, Alex Katz, Roy Lichtenstein, Kate Moss, Brad Pitt, Robert Rauschenberg, Lucas Samaras, and Philip Glass only serve to increase his demand as a portrait artist.
Certainly there are any number of other artists who do quite well financially. Jeff Koons and Damian Hirst may be foremost among these... but the prices of both artists seem greatly inflated through careful manipulation of the market (such as Hirst's purchase of his own infamous For the Love of God
... the diamond skull piece... for over $100 Million through a dummy consortium thus assuring himself great press and increased prices)... while neither artist has anything approaching a unanimous critical opinion. A great many of the best art critics, museums, and collectors dismiss their work out of hand as little more than a product of the time and the art market. We shall see. They do little for me.