I think Lou Reed qualifies, in the colloquial, if not technical sense (I don’t know if he had any other hits). Many people who only know him from the radio know him as someone who had one song, but he’s released well over thirty records, at least half of which I’d consider classics.
Did Ben Folds Five chart with any songs other than “Brick?” If not, they’d definitely fit the category.
Love and Rockets had only one hit in America, 1989’s “So Alive,” but their output in the 80s is well worth investing in (the first three albums being the best, the fourth being ok). Their 90s material is… not for everybody.
And even though I object to Thomas Dolby being lumped in the OHW category (Hyperactive, people! It counts as a hit… kinda… really!), his ability to avoid songs that suck is quite remarkable. Too bad he’s retired from the music biz.
Dean Friedman was a one-hit wonder in the '70s with Ariel. He never had another hit, but I enjoy his music, own two of his albums, and believe he is one of the finest songwriters I’ve heard. Infectuous melodies, mostly happy, Harry Chapin-type songs, not for everyone, Dean continues to record today, and has just released his latest album. He’s fluent in several musical styles, including jazz, swing, blues, and straight ahead rock, as well as the pop sound that got him his single top 30 hit.
Run, don’t walk, and buy every album Warren Zevon has made.
He only had one hit, “Werewolves of London”, and it’s not even close to being his best song. Not in the top 20.
Two words. Euro. Beat.
Haddaway was, in my mind, one of many fine singers who just never got his due. Go listen to Life Will Never Be The Same, or…well, anything. (Could be hard to find his albums now, though.)
Culture Beat has a ton of great stuff other than Mr. Vain. Most of it sounds pretty similar, too…can’t imagine why it didn’t catch on.
Likewise, La Bouche…I’m still mystified as to why Be My Lover was their only breakthrough hit. Sadly, with the tragic death of their female singer, any kind of future is in doubt.
Eiffel 65. Same argument. Why?
And of course, the #1 undisputed champ of all time, the band no list of hopelessly misrepresented, mismarketed, misunderstood, abso-friggin’-lutely doomed from day one bands is complete without…
Aqua. With, of course, their shocking megahit that sparked a lawsuit, gave them tons of negative press, and pretty much killed eurobeat in America stone-cold dead.
I advise everyone, regardless of musical tastes or whatever horrible memories they might have from that hideous song about a freaking lump of plastic, to get both their major albums, Aquarius and Aquarium (or download them if you have the means). Because while their other songs may not have poked fun at a corporate icon, they are perfectly representative of the light, bubblegummy dance music that made them such a hit in their homeland (Denmark. Or Norway…can never remember which.)
To this date, that one accursed song about the stupid, stupid doll which sparked a stupid, stupid lawsuit is the one Aqua song I cannot bear to listen to again. Ever.
Also:
Len - I guarantee you that just one listen of Cryptik Souls Crew will unlearn every impression That One Big Hit That Only Became A Hit Because Of That Razzafrakkin Movie Anyway gave you.
Crazy Town - They’re rapcore. Got it? RapCORE.
Midnight Oil - Was the fact that they were the Bad Religion of Australia, you know, too controversial for some people or something?
Martika - She had a voice that sounded like it was perfect for depressing, meaningless dirges. To this date, she still hasn’t gotten rid of that damn albatross…
Baha Men - Five…lousy…stinkin’…words…
Special mention for Enya for the unbelievably ham-fisted attempts to market her as a pop diva (which she never never never ever will be, get the freak over it already). Also, I wish everybody would stop treating Aretha Franklin and The Village People like one-hit wonders. (In case you’re wondering what I mean by that, think “endlessly repeated letters”.)
The lead singer from Aqua was Norwegian, but the rest of the band was Danish, and personally I’d say Denmark is welcome to claim the whole band 
But that’s not what I’m doing in this thread. You know what I’m doing in this thread, don’t you? Right. a-ha.
If you really liked “Take On Me”, I’d suggest giving Stay On These Roads a listen. It’s aged better than Hunting High and Low (the album not the song - the song is still one of my favorites), although it sounds like it was released before it was finished to me.
If you thought “Take On Me” was okay but liked “The Sun Always Shines On TV” much better, you are clearly a person of taste and refinement :D, and I’d recommend seeing if you can track down one or more of the following: Scoundrel Days, East of the Sun, West of the Moon", Memorial Beach, and Minor Earth Major Sky. Personally East of the Sun is still my favorite. Their newest, Lifelines, goes in a different direction and may not be what you’re expecting from a-ha.
(And if you were a teenager in 1985 and thought Mags was the cute one and not Morten, come on over sometime and we’ll swap hormones. I mean stories. ;))
Midnight Oil - “Beds Are Burning”, and so much more.
Check em out. I guess we won’t see any new stuff, since lead singer Peter Garrett has left the band. Bloody shame.
The Verve.
Their first full-length album, A Storm in Heaven, is in my Top 10 of all time.
The Eels got some radio play with “Novocaine for the Soul” and later, “Last Stop, This Town,” but they are such a great band. If you don’t like albums that are too depressing, I’d stay away from Electroshock Blues, at least at first, but all of their albums are really good.