In fact I can’t think of another modern popular song that is so well known that failed to make the charts. Perhaps one of you can think of one.
There are many -- Led Zeppelin's "Stairway to Heaven" is the poster child for this. It was never released as a single at all.
A few more big songs that either didn't chart at all on the U.S. Hot 100 or didn't make the U.S. Top 40:
Missed Hot 100:
"Crazy Train", Ozzy Osbourne (#106)
"What a Wonderful World", Louis Armstrong (#116)
"Blowin' in the Wind", Bob Dylan (n/a)
"Heroes", David Bowie (n/a)
"Comfortably Numb", Pink Floyd (n/a)
"Run Like Hell", Pink Floyd (n/a)
"Bad Reputation", Joan Jett
"Don't Go", Yaz (n/a)
"Blue Monday", New Order (n/a)
"Where's the Party?", Madonna (n/a)
"Into the Groove", Madonna (n/a)
"How Soon Is Now?", The Smiths (n/a)
"Back In Time", Huey Lewis and the News (n/a)
"Pink Cadillac", Bruce Springsteen (n/a)
"She Sells Sanctuary", The Cult (n/a)
"Never Say Goodbye", Bon Jovi (n/a)
"Orange Crush", R.E.M. (n/a)
"Higher Ground", Red Hot Chili Peppers (n/a)
Missed Top 40:
"Changes", David Bowie (#41)
"La Grange", ZZ Top (#41)
"Good Times Roll", The Cars (#41)
"Flirtin' With Disaster", Molly Hatchet (#42)
"Tom Sawyer", Rush (#44)
"Should I Stay or Should I Go?", The Clash (#45)
"Rebel Yell", Billy Idol (#46)
"What I Like About You", The Romantics (#46)
"Iron Man", Black Sabbath (#52)
"Rock Lobster", B-52s (#56)
"Sharp Dressed Man", ZZ Top (#56)
"Jam On It", Newcleus (#56)
"Hot for Teacher", Van Halen (#56)
"Surrender", Cheap Trick (#62)
"Purple Haze", Jimi Hendrix (#65)
"Rockit", Herbie Hancock (#71)
"Situation", Yaz (#72)
"Bizarre Love Triangle", New Order (didn't chart on original 1986 release, #98 upon 1995 re-release)