From Joan Acocella’s “Slaying Monsters,” in The New Yorker for 2 June 2014, a review of the recently published translation of Beowulf by J.R.R. Tolkien:

Tolkien was a serious philologist from the time he was a child. He and his cousin Mary had a private language, Nevbosh, and wrote limericks in it. One of their efforts went:

Dar fys ma vel gom co palt “Hoc
Pys go iskili far maino woc?
Pro si go fys do roc de
Do cat ym maino bocte
De volt fact soc ma taimful gyroc!”

(“There was an old man who said ‘How / Can I possibly carry my cow? / For if I were to ask it /To get in my basket / It would make such a terrible row!’ ”)

According to Michael D.C. Drout’s J.R.R. Tolkien Encyclopedia: Scholarship and Critical Assessment, New York: Routledge, 2007 p. 222:

Tolkien, with his cousin Mary Incledon, disguised French, English and Latin to create Nevbosh, or the New Nonsense, in which they wrote limericks.

It seems that this was not the first invented language Tolkien used since, according to this page which also provides a “Nevbosh Wordlist,” it came after Mary and Majorie Incledon’s Animalic, which he learnt.

The full story is told by Arne Zettersten in J.R.R. Tolkien’s Double Worlds and Creative Process: Language and Life. New York: Palgrave Macmillan, 2011, pp. 70-71:

The two cousins [Mary and Marjory Incledon] had already begun to construct a simple private language, which they called Animalic. Mary and Ronald continued to work on new inventions of words and the result was a completely new language, which was more advanced, and which they calle Nevbosh, or New Nonsense. Unfortunately, there is not much left of Nevbosh except a limerick, which is recorded in Tolkien’s essay, “A Secret Vice,” from 1931.
…
Tolkien points out in his essay that Nevbosh is mainly based on English, but that many words are changed or distorted. The creators of Nevbosh, Mary and Ronald, were children, without any conscious knowledge of the relations between sounds and concepts in language…
Tolkien also states in his essay that the dominance of English in Nevbosh might give the impression of being a “code.” He does not find this type of language interesting, but languages that are unrelated to traditional languages or school languages stimulate a deeper interest in him.

And on page 72 of the same book, in the context of a discussion of a picture letter, another Tolkien limerick is presented:

Tolkien also engaged in constructed alphabets during his school period. These had code symbols that corresponded to every letter in the english alphabet. A code letter from August 8, 1904 from Ronald to Father Francis survives in the Tolkien Collection at the Bodleian Library…
The letter ends with the following limerick to Father Francis:

There was an old priest named Francis
Who was fond of “cheefongy” dances
That he sat up too late
And worried his pate
Arranging these Frenchified Prances.