One of the “Limerick Company” I occasionally hear from, Bob Turvey, mentioned the following article from the 5 March 2023 Sunday Times “Culture” supplement, pp. 24-25:

Here is the passage of interest from the book being reviewed:

Another star, and arguably the most popular of all pre-mass production artists in America, was Russell Hunting, whose rapid-fire skits involving the Irish character Michael Casey were so popular they sparked an entire army of imitators. Hunting was an actor with the Boston Theatre Company who had leased a phonograph for his own use, just to experiment. Certain kinds of voice might come out as garbled scratchings, but while experimenting he realised he sounded particularly good on recond.
Hios debut record, with the New England Phonograph Company, came out in 1891. Soon, he was recording for Columbia and many others. Versions of his 1892 skits ‘Casey at the Telephone? and ?Casey Taking the Census? became his most famour works, as did the baseball poem ?Casey at the Bat’. Casey was so popylar that Hunting lost control as more and more recording artists imitated Casey themselves, in poor knock-offs of the origina. In a bid to stam out this annoying practice, he launched his own magazine, The Phonoscope, in 1896. …
Although he didn’t mention them in The Phonoscope, Hunting also put out a series of indecent recordings for saloons and amusement arcades on Coney Island and was briefly imprisoned for violating obscenity laws. He wasn’t alone either, and lots of cylinder nasties appeared in the 1890s, until legislation caught up and outlawed them. While most of these lewd recordings were destroyed, you can listen to examples that survived in the 2007 compilation Actionable Offenses: Indecent Phonograph Recordings from the 1890s. I’ve listened to them, so you don’t have to – it’s mainly profanity-laden skits and limericks.

You can read more on the CD on Wikipedia, and more on dirty recordings (of limericks) in There Once Was a Record of Smut by Jody Rosen from the New York Times website, which states:

His [an anonymous contributor to the CD] repertory includes “He’ll Win in a Walk, B’Jesus,” a scatological ballad about a day at the horse track; several dirty variations on “Mary Had a Little Lamb”; and off-color limericks like the one that begins “There was a young lady from Alaska.” (Let’s just say the punch line involves John Jacob Astor.)

Doug Harris, never one to miss an opportunity to discuss limericks, can’t seem to be able to find his copy of the CD, however he reports:

Interesting stuff… though it may be that only a single limerick (Alaska / Astor) has been recorded here and this might sadly make the claim; “mainly profanity-laden skits and limericks” to mean more of the former and less of the latter, perchance not even in the plural.

Unfortunately, I don’t know the whole “Alaska” limerick; however, more or less at the same time, I found the following in Lyndall Gordon’s The Hyacinth Girl: T.S. Eliot’s Hidden Muse:

Eliot once remarked that however intimate a love poem may be, it is meant to be overheard. An addition to his marital poems is a teasing limerick (in ‘Valerie’s Own Book’): ‘The Blameless Sister of Publicola’, dated 16 September 1959 and alluding to ‘The noble sister of Publicola’, Valeria, who is reported as icicle-pure in Coriolanus:

I know a nice girl named Valeria
Who has a delicious posterior
And beautiful thighs
Where her true lover lies
While his penis explores her interior.

Before their marriage, Eliot resisted – so he told Conrad Aiken – Valerie’s appeals to him to ‘burst into Bolovian song’. But afterwards he copied out assorted stanzas, previously reserved for men, in ‘Valerie’s Own Book’: twelve stanzas in one exercise book and two in the other.

The poem also appears in volume II of The Poems of T.S. Eliot, with other “Improper Rhymes.” The editors, Christopher Ricks an d Jim McCue, comment:

Valerie’s Own Book: fair copy, one page, dated “16. ix. 59”.

Coriolanus V iii, VOLUMNIA: “Do you know this lady?” CORIOLANUS: “The noble sister of Publicola; | The moon of Rome: chaste as the icicle | That’s curdied by the frost, from purest snow, | And hangs on Dian’s Temple: dear Valeria!”

While “his ‘King Bolo’ limericks” are not real five-line limericks, we find:

There was a young lady named Ransome
Who surrendered 5 times in a hansom,
When she said to her swain
He must do it again
He replied: “My name’s Simpson, not Samson”.

To Frank Morley, from Harvard, “St. Stephen Protomartyr [26 Dec] 1932”.

TSE had written to Morley, 20 Dec 1932: “I have an undeserved reputation for limericks which I must live down”, but less than a week later, enclosing this, he admitted “I know several good Limericks now.”

And:

There was a young girl of Siberia
Who had such a tempting posterior
That the Lapps and the Finns
Kept inventing new sins
As the recognised types were too stereo–.

Published in The Faber Book of Blue Verse from ms in Valerie’s Own Book. Date of composition unknown. C. L. Sulzberger recalled TSE at Harvard in 1933: “Timid and withdrawn as Eliot was in class, he had a talent for banging the piano and singing a huge number of limericks, some of which I suspect he had written himself”, A Long Row of Candles (1969) 4 (James Loucks, personal communication).

The introduction to the section also mentions the Eliots receiving Konrad Aiken’s collection of limericks:

Valerie Eliot to Aiken, 23 Nov 1964, acknowledging a copy of A Seizure of Limericks (1964): “Tom was so pleased to receive your limericks and we laugh over them together.” In an obituary for Life, 15 Jan 1965, Aiken wrote: “I’m the official custodian of his King Bolo poems, which are all quatrains or octets about an imaginary monarch and his queen, but he neglected to send the bulk of them to me.”

To conclude, it seems that one of the earliest poems Eliot wrote, in Fireside: A Weekly Magazine a family magazine he produced at the age of 10, was also a limerick, in no. 11, attributed to “Anon.” we find:

There was a young lady named Lu,
Who felt so exceedingly blue,
She was her-ad to state
That it was her fate—
And then she began to bu-hu.

Also of interest on Eliot’s bawdy poems: Jayme Stayer, “The Short and Surprisingly Private Life of King Bolo: Eliot’s Bawdy Poems and Their Audiences.” The T. S. Eliot Studies Annual, 2017, pp. 3-30, note 55:

On the topic of form and doggerel, the reader may wish to compare the three bawdy limericks that Eliot wrote out for others: “There was a young lady named Ransome,” “There was a young girl of Siberia,” and “The Blameless Sister of Publicola” (Poems 2, 286, 287, and 290). As would be expected of the oral tradition from which they come, these limericks involve some elements of repetition and variation. What matters, then, is not whether they are wholly original to Eliot, but to what extent the tight rhythmic and formal constraints of the limerick contrast with the looser, open-ended ballad form in which the Bolo poems were written. The last limerick plays on lines from Coriolanus and Valerie’s name, and is more likely original than the first. All three limericks are free of racist or misogynist content and are, in my estimation, more witty and delightfully obscene than all  of the Bolo poems and Bolo prose combined.