Edward Lear

Probably the best known limericist is Edward Lear, who in 1846 published his "First Book of Nonsense". He didn't call his verses limericks though they follow the form more or less precisely.
There are 212 of them and they follow a particular style not much used today, in that the 5th line is,  unlike modern limericks, less of a punchline and more of a recapitulation of the previous lines. So:

There was an Old Person in Gray,
Whose feelings were tinged with dismay;
She purchased two Parrots,
And fed them with Carrots,
Which pleased that Old Person in Gray.

The last line of this very well-known one is, however, more like today's form. The last few words of lines 1 and 5 are nearly always the same.

There was an Old Man with a beard,
Who said 'It is just as I feared! -
Two Owls and a Hen,
Four Larks and a Wren,
Have all built their nests in my beard!'

So lacking a conventional punchline, the last line is an anticlimax which gives the verses their particular attraction. Arguably his most famous poem is "The Owl and the Pussy-cat":

The Owl and the Pussy-cat went to sea
In a beautiful pea green boat,
They took some honey,
and plenty of money,
Wrapped up in a five pound note.

The Owl looked up to the stars above,
And sang to a small guitar,
'O lovely Pussy!
O Pussy my love,
What a beautiful Pussy you are,
You are,
You are!
What a beautiful Pussy you are!'

It's not strictly a limerick of course, and it has 2 more verses (Wikipedia has them). However I couldn't resist the thought of them spending the £5 and running out of money. Hence: 

Said the Owl to the Pussycat "I vote
That's it's time to sell the green boat
They'd have got a lot more
Than three pounds ninety-four
If the damned thing had managed to float

 Lear's books of nonsense poetry were popular in the 19th century, and encouraged many others to use the limerick form. Many modern limericks are rude; indeed there is a school of thought that holds that clean limericks are second-rate. Lear's limericks are never rude.