There’s another British counting-up song that’d fall in this category. I remember Hastings and Poirot sang it while driving through the countryside in an early David Suchet Poirot episode. It goes something like:
"One man and his dog,
went to mow, went to mow a meadow.
"One man, two men, and their dog
went to mow, went to mow a meadow.
“One man, two men, three men…”
And so on, adding more men each time.
“One naked man just walking down the road
Would cause a heap of trouble for all involved.”
Another one from Spain…
Un elefante
se balanceaba
sobre la tela de una araña,
como veía
que no se caía
fue a llamar a otro elefante.
Dos elefantes
se balanceaban…
One elephant
was swinging
on a spiderweb.
As he could see
he was not falling
he went and called another elephant.
Two elephants
were swinging…
Which ends when someone gets fed up and decides that the elephants do finally come crashing down.
Several of the songs by the so-called Payasos de la Tele (the Aragón brothers, who came from a long family of clowns; one of the most important showmen and producers in Spain is Emilio Aragón, previously known as Milikito, who is the eldest son of Emilio-called-Miliki and used to be on their show before breaking out on his own), while not having a counting element, are very good for trips as well. They usually include gestures, which makes them kind of hard to explain by just writing down.
Mi barba tiene tres pelos,
tres pelos tiene mi barba,
si no tuviera tres pelos
ya no sería una barba.
(Mi beard has three hairs,
three hairs are there in my beard,
if it wasn’t for those three hairs
it would not be a beard any longer)
trades one word for a gesture in each repetition, until it becomes
Mi [gesture as if you’re touching a ZZTop-style beard] tiene [raise three fingers] [touch hair],
[three fingers] [touch hair] tiene mi [touch beard]
si no tuviera [three fingers] [touch hair]
ya no sería una [touch beard].
You can find videos of these guys in YouTube. Look for “payasos de la tele” or “miliki”. And of course, now I have “Hola, Don Pepito” for an earworm 
Another from France which also has a B-I-N-G-O thing going on. (If you don’t know what I’m talking about, you’ll see in a moment.)
Napoléon avait cinq cent soldats.
Napoléon avait cinq cent soldats.
Napoléon avait cinq cent soldats.
–Marchez en même temps !
(Napoleon had five hundred soldiers (x3)
March in time !)
Then the second time you drop the last syllable from the Napoleon line :
Napoléon avait cinq cent sol— but no clap as you do with the Bingo song.
The last line (Marchez en même temps) is sung with full gusto.
The third time, you drop the last two syllables and successively down the line to the song’s conclusion.
Sur le pont d’Avignon (On the bridge of Avignon) – the cute, young student teacher who taught us the song in my high school French class was always able to pull out another “and the __ all ___.”
In German, we have the “Drei Chinesen mit dem Kontrabass” song for this.
Drei Chinesen mit dem Kontrabass,
sassen auf der Strasse und erzählten sich was.
Da kam die Polizei, “Ja was ist denn das?”
Drei Chinesen mit dem Kontrabass!
Three Chinese with the contrabass
sat on the street and told each other something.
Then the police came “Well, what’s all this then?”
Three Chinese with the contrabass!
After the first verse, all vowels are replaced with a, then with e and so on.
There’s a simple repeating song, like Henry the Eighth:
Ein Hund kam in die Küche,
und stahl dem Koch ein Ei.
Da nahm der Koch den Löffel,
und schlug den Hund entzwei.
Da kamen alle Hunde,
und gruben ihm ein Grab,
und setzten ihm nen Grabstein,
worauf geschrieben stand: “Ein Hund…”
A dog came into the kitchen and stole the cook an egg.
So the cook took a spoon and hit the dog into two.
So all dogs came and dug a grave,
and put up a gravestone, on which was written: “A dog came …”
Repeat until all adults are driven crazy.
There’s the “Die Wissenschaft hat festgestellt” (Science has found out) where the children invent stuff:
Die wissenschaft hat festgestellt, festgestellt, festgestellt,
dass Coca-cola Schnapps enthält, Schnaps enthält, Schnaps enthält.
Drum trinken wir auf jeder Reise, jeder Reise, jeder Reise
Coca-Cola eimerweise, eimerweise.
Science has found out
that Coca-Cola contains hard liquor.
Therefore we drink on every journey
Coca-cola by the bucket.
Also popular with many verses is “Wer hat die Kokosnuss geklaut”? (Who stole the coco nut)
Die Affen rasen durch den Wald,
der eine macht den anderen kalt,
die ganze Affenbande brüllt:
(kids start shouting) Wo ist die Kokosnuss, wo ist die Kokosnuss, wer hat die kokosnuss geklaut? (Repeat)
The monkeys are racing through the forest,
one is killing another,
the whole group is shouting
Where is the coco nut, who stole the coconut?
In a travel song book, there are about 10 verses, but you can also make up new ones.
Song with dropping words: the most common one is “Jetzt fahren wir übern See” (Now we travel across the lake)
Jetzt fahrn wir übern See, übern See, jetzt fahrn wir übern See
mit einer dicken Wurzel, Wurzel, Wurzel,
kein Ruder war nicht dran.
Now we travel across the lake
with a thick root,
no oar was not fastened to it.
On each repetition, you drop the words without making mistakes.
A popular song with gestures is “Ein kleiner Matrose” but that’s usually for Kindergarten children, not during travels.
Ein kleiner Matrose umsegelte die Welt.
Er liebte ein Mädchen, das hatte gar kein Geld.
Das Mädchen muss sterben, und wer war Schuld daran?
Der kleine Matrose in seinem Liebeswahn.
A small seaman sailed around the world.
He loved a girl, but had no money.
The girl had to die, and whose fault was that?
The small seaman in his love craziness.
Another one, with gestures, but also with silences, is “Mein Hut, der hat drei Ecken” (My hat has three corners) which wikipedia says is also known in English and Hebrew:
Mein Hut, der hat drei Ecken,
drei Ecken hat mein Hut.
Und hätt er nicht drei Ecken,
so wär er nicht mein Hut.
My hat has three corners
three corners has my hat.
And if it didn’t have three corners
it wouldn’t be my hat.
IME that’s always where the knives come out. . .
Missed the edit window. . . .
“Father Abraham” is one where the parts of the body are named and slapped as you go.
Father Abraham has many sons
Many sons has Father Abraham
I am one of them and so are you
So let;s all praise the Lord.
Right Arm! + Left Arm + Right Leg +Left Leg, adding one on each repetition. I’ve seen it get as far as femurs and phalanges and even an appendix once.
The balancing elephants and the three-cornered hat are translated into Swedish as well. I suspect in other languages as well.
Hola don José!
My mom taught us a counting song in French about crocodiles going off to war to fight the elephants. She’s from Quebec. It starts at one and counts upward. At this website (in French) it’s listed as a variant of a longer not-counting song.
The version I remember goes like this:
*
Un crocodile
S’en allait à la guerre
Disant adieu
À ses petits enfants
Sa longues queues
Trainaient dans la poussière
Il s’en allait
Combattr’ les éléphants
Si les cro-co-co les cro-co-co les crocodiles
Sur les bords du Nil se sont perdus n’en parlons plus !*
And so on…
Uh… French speakers, ignore my conjugating mistakes.
Anyway, I meant to add that the crocodile song is kind of for the same age group as “Alice the Camel”.
Hey, he had more in Spanish!
Na-po-león tenía miiiiiil ¡soldaos!
Na-po-león tenía miiiiiil ¡soldaos!
Na-po-león tenía miiiiiil ¡soldaos!
Marchando al rantamplán.
(Napoleon had one thousand soldiers
following the drums)
So tell me, Frodo, ¿pasó usted por mi casa?
One man and his pet Pomeranian poodle pup, Spot!
I’ve always heard it end with Turn around as the penultimate, and after that verse, “sit down” is appended, ending the song.
…nod your head, turn around, SIT DOWN!
It goes:
One man went to mow,
Went to mow a meadow,
One man and his dog,
Went to mow a meadow.
Two men went to mow,
Went to mow a meadow,
Two men, one man and his dog,
Went to mow a meadow.
And so ad infinitum (that third line gets difficult to do in one breath)