Little Eva: 1962
Grand Funk: 1975
Kylie Minogue: 1988
When that happened, I was not the only person who predicted that it would hit #1 a fourth time, by a different artist, in 2001. Didn’t happen, although it’s not too late for this year. Anyone? 
The Kylie Minogue version only reached #3. (It was a #1 in her native Australia, though.)
Well, no.
Your first three sentences are right on the money. However, “Without You” was written in 1970. Pete Ham committed suicide in 1975.
And it was actually a combination of two separate songs that originally had different titles. Its verse comes from Pete Ham, while the chorus was written by Badfinger bassist Tom Evans. The lead singer of each section is its composer. So the passion you’re hearing in the chorus is actually Tom’s, not Pete’s.
The albums “Rubber Soul” by the Beatles and “Love Song” by Anne Murray both got into the top 40 (#1 and 24, respectively) and had the song “You Won’t See Me” on them. Does that count?
No. Again, the discussion is about singles that made the Billboard Top 40. “You Won’t See Me” was never a Beatles single.
Well Andrea Bocelli released ***Con ti Partiro ***in 1995,on his second album “Bocelli” and it blew the doors off of the cross over charts in Europe, then in 96 Sarah Brightman recorded the song with Andrea featured as a duet on her album Timeless, and it was a hit world wide notably platinum in Us and Canada, 35 weeks #1 on the US billboard cross over chart. The song was played on the top 40 stations as well.
Brightman later recorded it in English as solo artist. Interestingly the solo recordings did well for the demographic, but the duet recording was an international hit.
They also both had hits with “Don’t Play That Song” (with both versions peaking at #11)
Some more…
“Release Me” Esther Phillips (1962)/Englebert Humperdinck (1967)
“C.C. Rider” Chuck Willis (1957)/LaVern Baker (1963)
“All By Myself” Eric Carmen (1976)/Celine Dion (1997)
“More Love” Smokey Robinson & the Miracles (1967)/Kim Carnes (1980)
“You’ve Made Me So Very Happy” Brenda Holloway (1967)/Blood, Sweat & Tears (1969)
“Where Did Our Love Go?” The Supremes (1964)/Donnie Elbert (1971)
“There’s a Kind of Hush” Herman’s Hermits (1967)/The Carpenters (1976)
“Do You Want To Dance” Bobby Freeman (1958)/Bette Midler (1972)
Borderline case:
“Respect Yourself” The Staple Singers (1971)/Bruce Willis (1987)
Borderline since (a) Pops Staples sings several lines on the original; (b) ‘backup’ singer Bonnie Pointer is more prominent than Willis himself on the remake.
Blue Moon was done by Lena Horn and the Marcels.
“You Can’t Hurry Love” by the Supremes reached #1 in the U.S. and the U.K. in 1966, then was recorded in 1982 by Phil Collins and reached #10 in the U.S. and #1 in the U.K.
“Never Can Say Goodbye”, Jackson 5, Gloria Gaynor
“Dream a Little Dream of Me”, Frankie Laine, Jack Owens, Cass Elliot
Oh, wow. Thanks to everyone for their responses.
One that hasn’t been mentioned is one of the most covered songs on the planet–Memory from Cats. Elaine Paige’s original version chartered, then later Barry Manilow’s was a hit.
As an aside, I’ll never forget the Streisand bio that stated “Elaine Paige’s cover of Streisand’s original version…” I think the author should have checked who covered whom.
Once again, this is not so according to the official definition of “Top 40.” Barry Manilow’s version did make the Top 40 (barely, at #39). The Elaine Paige version did not.
Additional Top 40 hits for Ronstadt that qualify:
“Tracks of My Tears” and “Ooh Baby Baby” (Smokey Robinson & the Miracles)
“Tumbling Dice” (The Rolling Stones)
“Back in the USA” (Chuck Berry)
“Hurt So Bad” (Little Anthony & the Imperials)
Well, it did in Europe.
While it was noted that Kristoferson’s version never charted, Jerry Lee Lewis recorded a cover that reached #40, so the song does (barely) qualify.
Additional examples…
“Mack the Knife” Bobby Darin (1959)/Ella Fitzgerald (1960)
“(If Loving You Is Wrong) I Don’t Want to Be Right” Luther Ingram (1971)/Barbara Mandrell (1979)
“Tell It Like It Is” Aaron Neville (1966)/Heart (1983)
“Eleanor Rigby” The Beatles (1966)/Aretha Franklin (1969)
“I Want to Take You Higher” Sly and the Family Stone (1969)/Ike and Tina Turner (1970)
“Let’s Stay Together” Al Green (1971)/Tina Turner (1984)
“I’m Into Something Good” Earl-Jean (1964)/Herman’s Hermits (1964)
“Sincerely” The Moonglows (1955)/The McGuire Sisters (1955)
“Don’t Leave Me This Way” Thelma Houston (1977)/The Communards (1987)
“Can’t Take My Eyes Off Of You” Frankie Valli (1967)/Lauryn Hill (1998)
“Our Day Will Come” Ruby and the Romantics (1963)/Frankie Valli (1975)
“Will You Love Me Tomorrow” The Shirelles (1961)/The Four Seasons (1968)
“Walk On By” Dionne Warwick (1964)/Isaac Hayes (1969)
“You’ll Never Get To Heaven” Dionne Warwick (1964)/The Stylistics (1973)
“I Only Want To Be With You” Dusty Springfield (1964)/The Bay City Rollers (1976)
“Cry Me A River” Julie London (1955)/Joe Cocker (1970)
“Love on a Two-Way Street” The Moments (1970)/Stacy Lattisaw (1981)
“You’ll Lose a Good Thing” Barbara Lynn (1962)/Freddy Fender (1976)
“All I Really Want To Do” The Byrds (1965)/Cher (1965)**
“Too Many Fish in the Sea” The Marvelettes (1965)/Mitch Ryder & the Detroit Wheels (1967)**
“Unforgettable” Nat ‘King’ Cole (1952)/Dinah Washington (1959)
And Floyd Cramer’s piano instrumental “Last Date” (1960) had lyrics added to it to become Skeeter Davis’ “My Last Date (With You)” (1961)
*Along with Louis Armstrong (1956) and a bunch of others
**Bob Dylan’s original did not chart
***As part of a medley with “Three Little Fishes”