The only one I remember is the start of a bawdy song…
When a man grows old, and his balls go cold
And the tip of his prick turns blue,
And it bends in the middle like a one-string fiddle
He can tell you a tale or two!
The girl I brought home was a prize
With beautiful deep blue eyes
Her tits were well kept
Just as you’d expect
But her penis came as a surprise
::my my, what an educated shower we are:: 
To his club-footed child said Lord Stipple
As he poured his post-prandial tipple
“Your mother’s behavior
Gave pain to our Saviour
And that’s why He made you a cripple.”
There was a young curate whose brain
Was deranged from the use of cocaine.
He lured a small child
To a copse dark and wild,
Where he beat it to death with his cane.
The first child of Mrs. Bysshe-Shelley
Came to light with it’s face in it’s belly.
The second was born
With a hoof and a horn.
And the third was shapeless as jelly.
My favorite is not raunchy, but quite unique:
1,264,853,971.2758463
[spoiler]Read as:
One billion two-hundred and sixty
Four million eight hundred and fifty
Three thousand nine hun-
dred and seventy one
Point two seven five eight four six three[/spoiler]
There once was a fellow from Yuma,
Who told an elephant joke to a puma,
Now his skeleton lies,
Under hot desert skies,
For the puma had no sense of hu-ma.
There once was a man named Bright
Who traveled much faster than light.
He left one day
in a relative way,
And came back the previous night.
While Titian was mixing rose-madder
His model posed nude on a ladder;
Her position, to Titian,
Suggested coition,
So he climed up the ladder and had 'er.
A young down-east pilot named Sanger
Got a girl alone in his hangar
When she asked where in Maine
He was flying his plane
He said he was going to Bangor
On the chest of a barmaid from Vail
Was tattooed all the prices of ale
Whilst on her behind
For the sake of the blind
Was precisely the same, but in braille.
Variation on that theme:
There was a young lady of Wheeling
A gal of most delicate feeling
When she read on the door,
“Don’t piss* on the floor”,
She jumped up and pissed* on the ceiling!
- or (as the bowdlerized version I originally learned had it) “spit” and “spat”, respectively
There was a young girl named Alice
Who used dynamite instead of a phallus
And when she lit a match
They found half of her snatch
In Ft. Worth and the rest in Dallas
Of course I meant Jill’s name was changed to Alice.
This is a fairly tame one -
There was an old woman named May,
Who once went to a park by the bay.
She met a young man,
Who screwed her and ran,
Now she goes to the park every day.
And this one doesn’t quite fit the standard rhyme scheme for a limerick, but -
Molly Brown, Molly Brown,
No one claimed to lay her down.
Then over the hills came piss-pot Pete
With 20 pounds of swinging meat.
Pete fucked Molly in the ass,
Molly passed one hell of a gas.
Then over the hills flew piss-pot Pete,
With 20 pounds of shredded meat.
There once was a man from Madras
Who’s balls were made out of brass
He’d sing “Stormy Weather”
As he rubbed them together
Whilst lightning flew from his ass.
And the variation on the theme already mentioned several times:
There once was a young lady named Alice
Who tried out a dynamite stick for a phallus
They found her vagina in North Carolina,
And parts of her anus in Dallas
There once was a girl from the Cape
Who loved a recalcitrant ape.
Her attempts at romance
Didn’t get in his pants,
And so she resorted to rape.
*There were two young ladies from Birmingham,
And this is the story concerning 'em,
That they pulled up the frock
And pulled on the cock
Of the Bishop as he was confirming 'em.
But the Bishop was nobody’s fool,
He had been to parochial school,
So he pulled down his britches
And diddled those bitches
With a twelve-inch Episcopal tool.
But both the young ladies said, “Pooh!”
And they sneered as the Bishop withdrew,
And they said, “Well, the vicar
Is longer and thicker
And slicker and quicker than you.”
*
Or
*Once a young gay from Khartoum,
Took a lesbian up to his room.
But they argued and fought
About who should do what
And how and with which and to whom.
*
Regards,
Shodan
This one won a contest. (I didn’t write it, FWIW.) It’s a bit dated, but it stuck with me (as limericks tend to do.) The contest was to write limerick that included “Clinton,” “Lewinsky” and “Kazinsky.”
Clinton and Lewinsky have shown
What Kazinsky must surely have known
That an intern is better
Than a bomb in a letter
Given the choice of how to be blown
When I was about eight I bought a Halloween-themed record from Scholastic Press that had a similar one:
A skeleton once, in Khartoum
Invited a ghost up into his room
They spent the whole night
In the eeriest fight
Over who should be frightened of whom
The first one I ever heard:
A wonderful bird is the pelican;
His beak can hold more than his belican;
Within that great beak,
He holds food for a weak,
But I’m damned if I know how the helican.
Google Eskimo Nell for the rest of that one. 
Much too long to post here.