Michael Bloomberg was elected mayor five years ago after a self-financed campaign in which he shunned the politician label.
A self-financed campaign by Saul could pose a formidable threat to the freshman incumbent.
The system has not been helped by candidates who opt out, like Mr. Bloomberg, whose self-financed campaigns cost an estimated $150 million.
Mr. Bloomberg ran two self-financed campaigns and declined to participate in the program - refusing matching funds and also exempting himself from spending limits.
Businessman Tom Knox had started off strong as his self-financed campaign created a barrage of advertising.
Mr. Edwards's self-financed campaign drew little national attention.
Hart received 26 percent of the vote (second place) in a largely self-financed campaign.
Schroeder spent $2.1 million on a largely self-financed campaign for Governor of New Jersey in 2005.
Republicans, meanwhile, said they were confident they could use Ms. Cantwell's largely self-financed campaign, and her short career in Congress, against her.
Also, attorney Phil Preis entered the race as a Democrat, and with a self-financed campaign was able to enter the top tier of candidates.