Does humuhumunukunukuapua’a count? It’s not nonsense, but it’s inexplicable unless you know what it is.
I think that fits under this particular umbrella in that:
- it’s weird enough
- you provided a link
- I know that I an happy to know about that particular nameable thing that I didn’t know before, and I hope others are as well.
These are definitely weird and the OP only said perhaps untranslatable. So I submit them in honor of my father, who introduced me to all of them.
Mairzy doats and dozy doats and liddle lamzy divey A kiddley divey too, wooden shoe?
Along the same lines:
*Anaal nathra uthe vas bethod doch yel Dienve *
Outstanding! You beat me to those first two, and I was just waiting to let some more water flow over the dam. That last one reminds me of similar stuff my own dad either dreamed up or got from some source I have yet to find! 
I can’t believe no one has mentioned Supercalifragilisticexpialidocious yet.
Yeah, Dads. Your reference to the dam made think of the Three Little Fishies, with:
Boop boop dit-tem dat-tem what-tem Chu!
Someone else is going to have to post a good link, though. Most of the lyrics sites are blocked, here.
I remember at least one version of the song that lisped through the lyics in a way to make the verses a bit hard to parse. If the chorus means anything, I’ve never heard what.
Kind of a long shot, but…
[Shool! Shool! Shool! I rule!
Shool! Shool! Shool! I rule!
Shool! Shacker-rack!
I shool bubba cool.
Seller! Beller eel!
Fust to ma tree’l
Just came er bubba.
Buska! Buska-reel!](The Project Gutenberg e-Book of Negro Folk Rhymes, by Thomas W. Talley.)
If you know it, I’d love to hear about it.
ETA: You might know it better as “Ish come bibble ahly-boo-so-real”
Gliddy gloop gloopy naba nibby noopy la la la lo lo
Saba sibby saba, nooby abba naba, lee lee lee lo lo
Tooby ooby walla, nooby alla naba,
Early mornin’ singing song
Good morning, Starshine!
Elroy!
Mekka lekka hi, mekka hiney ho
Oddly, I thought of the phrase immediately, though it took me 20 seconds or so to realize where it was from…