If we are just talking USA here, I’m going to put forth Creedence Clearwater Revival as a possible answer. Lots of Top 10, but no #1 on the Billboard chart.
Springsteen was going to be my guess as well. The link above says 12 top 10 hits (American) with 0 number ones (The 11 #1s is #1 albums, not songs).
CCR is the highest runner up I’ve found so far, with 9 top 10 hits.
Wikipedia has complete discographies for all major artists. You can check highest chart position for singles and count top tens. I Googled for acts with no number ones first.
Been ninjaed while I was playing with names.
Creedence Clearwater Revival has 9.
Bruce Springsteen has 12.
Missy Elliott has 5 but an additional 6 as featured artist.
You can monkey with the Billboard link above by entering the artist name with hyphens for spaces into the URL and the top line will give you a summary count of #1s, Top 10 and Top 100 US chart singles.
Other runner ups are ELO with 7, and surprisingly Journey with 6. I think every dance when I was in high school was required by law to play “Open Arms”, which peaked at #2.
If this type of music topic interests you, I could not recommend “Hit Parade” podcast more highly.
It’s by a music historian, Chris Molanphy who basically looks at the stories behind billboard hits. He deals with both bands and eras of music.
To answer your original post I believe he does concur that it is CCR.
In just a couple of years, Creedence Clearwater Revival generated one of the most amazing runs of hits in American pop history: from “Proud Mary” to “Green River,” “Bad Moon Rising” to “Travelin’ Band.” Reportedly, they even outsold the Beatles in America in 1969. But for all their success with those John Fogerty–penned classics, CCR never held the No. 1 spot on the Hot 100. All of those hits were No. 2s: a Billboard chart record they hold to this day, for most No. 2s without a No. 1.
I was curious I went through my spreadsheet of favorite songs. I tried to do an equivalent of top 10 without being a number one.
My top 2 is a tie with Rolling Stones and Jim Croce with 9. Third place with 8 was Bruce Springsteen.