What is the earliest known limerick?

Edward Lear wasn't the first, but he made it popular.

https://en.wikipedia.org/wiki/Edward_Lear

So far as may be verified, the oldest of these are also found among the "Mother Goose" rhymes, collected by Halliwell.

(Halliwell's collection of English Nursery Rhymes, among a large mass of jingling folk-lore, to which it is impossible to ascribe definite dates, but which was current about the fifteenth or sixteenth century.)

There was an old man of Tobago,
Who lived upon rice, gruel and sago;
Till, much to his bliss,
His physician said this:
"To a leg, sir, of mutton, you may go."

There was an old soldier of Bister,
Went walking one day with his sister;
When a cow, at one poke,
Tossed her into an oak,
Before the old gentleman missed her.

https://nonsenselit.wordpress.com/limericks-1903/

Variants of the form of poetry referred to as Limerick poems can be traced back to the fourteenth century English history. Limericks were used in Nursery Rhymes and other poems for children. But as limericks were short, relatively easy to compose and bawdy or sexual in nature they were often repeated by beggars or the working classes in the British pubs and taverns of the fifteenth, sixteenth and seventh centuries. The poets who created these limericks were therefore often drunkards! Limericks were also referred to as dirty.

http://www.poetry-online.org/limericks.htm

A limerick is a five-line poem with a set form. The rhyme scheme is AABBA. The third and fourth lines (the 'B' lines) are shorter than the rest of the lines.

Traditionally, limericks tend toward the bawdy and the tawdry. However, early in the life of the form, limericks were as benign as other forms of poetry. For example, you'll probably recognize this 1774 limerick:

Hickory Dickory Dock
A mouse ran up the clock
The clock struck one
And down he run
Hickory Dickory Dock.

Limericks began to gain their widespread popularity in the mid-to-late eighteen-hundreds with the publication of Edward Lear’s Book of Nonsense in 1845 and 1872. His verses centered on nonsensical themes and often utilized wordplay. Although they were usually printed in 3 or 4 lines to accommodate illustrations, many of his verses use the limerick rhythm and rhyme scheme we recognize today:

There was an old person whose habits Induced him to feed upon rabits
When he'd eaten eighteen he turned perfectly green
Upon which he relinquished those habits
The word limerick first came into use at the end of the nineteenth century.

http://www.bellaonline.com/articles/art20882.asp