Maurice Williams and the Zodiacs - Stay.
The shortest Billboard #1 of all time, clocking in at about 90 seconds.
Actually, this one might make the list because the title fits perfectly, I know it’s a song they’d play, and Robert Cray played a few years ago at the festival I’m trying to get the tickets to. Because so many begging/pleading songs are about love. and what I’m going to poke fun at in the letter is how one of their DJ’s begged on-air for tickets to another event, I think the list is going to have to go on the merits of the song titles.
I like the Please, Please, Please suggestion too but I don’t think I’ve ever heard that song myself and don’t know if they’ll have it in their library.
Thanks for all the suggestions so far and keep 'em coming!
As I mentioned in my Song ID thread I tracked down the song I was trying to think of in my OP. It’s “Ticket” by Sherri Jackson (who sounds nothing like Suzanne Vega).
“Please Please Please Let Me Get What I Want” by The Smiths. Or Morrissey, I forget which.
Pretty much anything by Boyz II Men would do it. 
It’s the Smiths.
“Please Don’t Go” by KC and the Sunshine Band.
John Lennon also did a song called Please Don’t Go.
Led Zeppelin’s “D’yer Mak’er”
Oh, oh, oh, oh, oh, oh,
You don’t have to go
Oh, oh, oh, oh, oh…
I’m not sure if the criteria for your contest require actual use of the words “beg” or “plead”, but here’s some suggestions that fulfill the spirit, if not the letter: “I Can’t Help Myself”/Four Tops, “When A Man Loves a Woman”/Percy Sledge, “Sexual Healing”/Marvin Gaye, “Slip Away”/Clarence Carter, “Can’t Get Next to You”/Al Green, and “I’m Your Puppet”/Purify Brothers. All of the above except the last (AND the wonderful Otis Redding cut suggested above by Marley23) were on a courtin’ mix tape I sent to my long-distance girlfriend… some of the greatest voices in music playing Cyrano for me.
She’s now my wife. Heh! I’m such a playaaa! And a lucky, lucky man.
Bruce Springsteen: “Rosalita”