By starred, I mean top billing, and by successful, I mean a show that lasted at least three seasons.
I know David Boreanaz has had three: Angel, Bones, and now Seal Team. Would he be the one with the most?
Definitely not David Boreanaz. Ted Danson, for example, has been on Cheers, Becker, CSI (not original cast, but for more than three seasons), Damages, Bored to Death and The Good Place.
And I’ll bet someone can name someone more successful than that.
Lucille Ball starred in four shows that went over 3 seasons: I Love Lucy, The Lucy-Desi Comedy Hour, The Lucy Show, and Here’s Lucy.
We asked this years ago and the answer was Bill Bixby, who had had multiple shows go multiple seasons(I forget how many).
For the OP’s criteria, top billing and at least 3 seasons, I think Bill Bixby only has three: My Favorite Martian, The Courtship of Eddie’s Father, and The Incredible Hulk.
Also, I don’t think that Boreanaz actually had top billing for Bones; I think that may have gone to Emily Deschanel.
He shared top billing. He was clearly a co-star at the very least. Lucy doesn’t count, it’s all the same show. Danson, however, I think by the criteria I set out he’s the best so far.
For Ted Danson, I think Cheers, Becker, and The Good Place would all definitely count per the OP’s criteria, but he was only a series regular on Damages for the first season, and I’m not sure he counts as getting “top billing” for either CSI or Bored to Death. Still, that makes him at least tied with Boreanaz, and, depending on how you define “top billing”, he may beat out Lucille Ball as well.
Sounds like there’s a dispute for one of Boreanaz’s three and one of Danson’s three, but Scott Bakula was the straight-up star on QUANTUM LEAP, and is doing likewise now on NCIS: NEW ORLEANS, and in between was the captain of the ENTERPRISE.
Why doesn’t Lucille Ball count? The four shows I listed are not the same show, they just have similar names.
I Love Lucy and The Lucy-Desi Comedy Hour might arguably have been “the same show,” as they featured the same actors, playing the same characters.
But, The Lucy Show and Here’s Lucy were different series entirely, with her playing different characters (even though Vivian Vance (Ethel Mertz from the earlier series) was in The Lucy Show, and Gale Gordon was in both of the later series).
Harry Morgan: December Bride, Dragnet, and Mash. Pete and Gladys ran two seasons, so that’s close.
Bea Benaderet: Burns and Allen, The Flintstones, Petticoat Junction
John Forsythe: Bachelor Father, Charlie’s Angels, Dynasty
William Shatner : Star Trek (TOS), TJ Hooker, Boston Legal
William Shatner has Star Trek, T.J. Hooker, Boston Legal, and arguably Rescue 911. (ETA: Ninja’d!)
I don’t see Harry Morgan as starring in December Bride, nor Bea Benaderet as a star of Burns and Allen.
OK, Mr. Smartie Pants, who did have the top billing in Burns and Allen?
If you’re going to include Rescue 911 (18 episodes over 7 seasons), then you need to include TekWar (18 episodes over three years but ONE season 
)
Or, maybe not.
Shatner rules TV!
Helen Mirren starred in Prime Suspect, Elizabeth I, and in Catherine the Great.
I don’t think Elizabeth I or Catherine the Great lasted long enough to qualify, given the restrictions in the OP.
1558 to 1603 and 1762 to 1796 aren’t long enough for you?