Thanks for asking such a great question, which really made me think! Hadn't realised how many great logos there are out there, or how much we take them for granted, until I started trying to choose my favourite. Then I realised I was spoilt for choice.
In terms of ones which have a story behind them - and real character - I think I'm going to pick the Michelin man, also known as "Bibendum", or "Bib", or "Bibelobis".
Here he is, in his current form - though I much prefer some of the older versions:
Bibendum is one of the world's oldest trademarks, being introduced in 1894 at the Lyon Exhibition.
His slogan, "Nunc est bibendum" means "Now is the time to drink", and is a quotation from The Odes, by Horace, a Roman lyric poet.(Source: http://logoblink.com/michelin-logo-in-vintage-posters/)
Apparently, the Michelin brothers noticed that a pile of tyres (or tires, if you're in the US) resembled a male figure and, some years later, when they met French cartoonist, Marius Rossillon, they mentioned this.
Rossillon (also known as O'Galop) then showed them an image he had created for a client, who had rejected it. This showed a man holding a glass of beer in the air, and included the quote from Horace.
The Michelin brothers suggested replacing the man with one made from bicycle tyres, and so Bibendum was born.(Source: http://violetsvintagevault.wordpress.com/2012/08/)
He hasn't remained unchanged over the years, though - and has been modified to suit changing tastes and cultural mores.
Now, he has become slimmer, has given up his cigars, and also his pince-nez, or glasses. I preferred him as he was originally!(Source: http://www.designclarksville.com/whaddya-know-michelin-man-evolution/)
I think the reason that I like this logo so much is because it has such a sense of humour, as well as a distinct personality of its own. It's irreverent, instantly-recognisable, and the phrase, "Michelin Man" has even entered the popular vernacular.
I say it about myself, when I'm layered up in thermal underwear, in the winter - and, funnily enough, "Bib" happens to be my son's nickname, amongst the family, though that's entirely coincidental.
If anyone wants to add their favourite logo to the answers to this question, there's a great selection of famous ones here.