For a song with a bass line with that quality, try Talking Heads “Once in a Lifetime.” (The entire song, except for an initial E, is a repeated two bar phrase consisting of two different notes: F# and A. I’m not a bassist, but it is actually a bit tricky for me to retain that level of concentration through the whole song with its different verse, chorus, and bridge parts. Pretty clever.)
Here are my suggestions in no particular order or genre.
Sublime or any Ska based music thinking “Wrong Way” as a start.
The Dead South. “In Hell I’ll Be In Good Company”
The Band “The Weight”
Metallica “For Whom The Bells Toll” the beginning solo.
As for suggestions to the OP, how about Carol Kaye’s bass line for “Good Vibrations”? Fun rhythmic parts in the verse; cool walking type thing going on the chorus. Nice stuff.
I second Dinsdale’s Motown suggestion. So much bass fun in those pieces. “It’s the Same Old Song” or “I Can’t Help Myself” by the Four Tops, “Aint Too Proud to Beg” or “Get Ready” by the Temptations, “I Want You Back,” “ABC,” “Darling Dear,” by the Jackson 5 (already mentioned–but those are trickier.) Lots of great bass all over Motown.
I love old rock opera stuff from Broadway. Stuff like “Hair” and “Jesus Christ Superstar”
Those bass lines are fast moving and tend to do a whole lot with the middle range of the instrument. I like that because it is a different sound from always reaching for the low notes, and you can almost always play those songs with a 4 string.
One of my favorites is the opener from JCS, Heaven on Their Minds.
It start a with a driving beat that carries on for the first third of the song, it shifts gears into a neat backing line that uses plenty of major and minor pentatonic stuff.
The full score for JCS is well over 200 pages long, and they were not slacking for any part of it. Those musicians sure earned their pay.
I have messed around with the bass for many decades. I can’t read music, nor do I want to learn. I also don’t care for tablatures. I get the most enjoyment from trying to figure out a song’s bass line by listening to it over and over and trying to figure out each note by ear.
Some stuff is just about impossible for me to learn “by ear,” either because it’s too complex (e.g. Zappa, Joni Mitchell), or I simply can’t hear the bass very well. OTOH, I have found the bass lines by some groups/musicians are pretty easy to learn by ear.
When I was a teen I figured out the bass lines for many of the songs on Black Sabbath’s first four albums. Most are not difficult to figure out. And I love the style.
I know the OP said he didn’t care for them, but if you are new to the bass, most songs by The Ramones are easy to figure out. I love playing along with them. (On The Ramones’ first album, turn the balance control all the way to the left channel. The only guitar you will hear is Dee Dee’s bass! Super easy to figure out the bass lines when you do this.)
Probably the “funnest” album I have ever figure out the bass lines on is Big Lizard in My Backyard by The Dead Milkmen. It’s a blast to play along with; the bass is right out front (so you can clearly hear it) and the songs are hilarious.
Thanks to everyone! There are some great and some hilarious suggestions here. bordelond, sure I’ll get right on those. I might actually be able to play parts of the Duran Duran song, though – that will go on the list, since that video lays it out so well. Crafter_Man, it’s not that I don’t like the Ramones, it’s just that their bass parts are pretty mindless. minor7flat5, no iOS devices, unfortunately. I do use Rocksmith on the PC, which has helped get my fingers moving on the neck and strings, and there are hundreds of songs on YouTube that use the same interface.
I’m on vacation, so I can’t respond to every post, but I really appreciate the responses. When I get back home, I’ll definitely try all of the reasonable ones here. I’m not trying Primus – that guy is a bass (the instrument, not the voice part) god.
I guess I can’t disagree with that. But I still love it. 
I would like to reiterate just how wonderful the bass lines are on Big Lizard in My Backyard by The Dead Milkmen. As an example, check out Beach Song. Should only take you a few minutes to figure it out. And other than the chorus - which is even easier - that’s the whole song! Same goes for Bitchin’ Camaro. For something a bit more challenging, figure out the bass line in Swordfish. It sounds complicated, but really isn’t.
You missed the first minute of it. I was referring to the guitar+bass riff at the beginning, not to the abstract atonal jam that followed. That guitar+bass riff is also reprised about a minute from the end. I wasn’t suggesting you mess with the stuff in the middle!
When I was young some friends of mine who couldn’t read music and could only play open-tuned guitar chords with a single finger would jam together. They mostly played Hawkwind songs, but I first learned “Interstellar Overdrive” from jamming with them. The riff starts on B and descends to low E based on an elaboration of the chromatic scale. I recommended it as good figure-it-out material.
That was Grand Funk Railroad eh.
:smack:
You’re right, of course.
Have they and The Guess Who ever appeared on the same stage at the same time? Hmmm? 
My favorite: Gerry Rafferty’s “Right Down the Line”
It’s not too hard to figure out, but it does require some dexterity.
I definitely nailed that part! This guy fixes it: https://www.youtube.com/watch?v=7ORinrCH7oM
(Just coming back and trying out some of these suggestions – thanks, all!)
Glad to see this thread pop back up; I thought of someone else you might wanna look into.
Maybe not popular enough (if you’re trying to find songs you can play with other people, I mean), but solid, fun, easy-to-hear bass parts can be found on Nomeansno’s Wrong album.
Kick-ass band, just not super-well-known.
Funky!
“It’s my life,” by Talk Talk is kind of fun.
Hehehe, they are easy to hear (and he’s one of my heroes), but I wouldn’t necessarily call the parts on Wrong easy. It’s an exercise!
Dammit, that album kills, though. That bastard sang while playing those riffs. All hail Rob Wright, the BASS ICE GIANT OF THE NORTH!
Takin Care of Business - Bachman–Turner Overdrive
Let It Ride
You Ain’t Seen Nothin Yet
BTO has fun bass lines. Fred Turner plays bass pretty well. Especially considering that he provides vocals too.
The bass is easy to hear in this live footage of Let It Ride.
You can see Fred’s hands too. 
Fred Turner’s growl in the vocals is so cool. He’s thumping that bass the whole time.
I will, of course, defer to your expertise and experience. To my ear, they sound active but not complicated. Lots of eighth notes, fer sher. And hey, any time I can help people discover music is a big, big bonus to me, so I try.