The most famous one will probably be Rocky Raccoon from the White Album, but there are many more (and Ringo would later go on to record a whole bunch of them on Beaucoup of Blues).
Here are some that you might consider listening to:
- Act Naturally: a Ringo cover of a Buck Owens country song
- What Goes On (the only song credited to Lennon-McCartney-Starkey)
- Don't Pass Me By (again Ringo)
- I'll Cry instead
- Many songs on Beatles for Sale - I've Just Seen a Face: a Paul McCartney song that was influenced by Country & Western. In an interview Paul says: "It was slightly country and western from my point of view... it was faster, though, it was a strange uptempo thing". If it had been slowed down a bit, it would have been pure country. And if you had added fiddle and banjo, you might even have had a bluegrass or cajun song!
p.s. why was Ringo so fascinated with Country & Western?
Well, first of all, he watched a lot of cowboy movies and read cowboy comics when he was young. Allegedly the name Ringo comes from a character (Johnny Ringo) that took part at the Gunfight at O.K. Corral.
Later on Ringo joined a band called The Raving Texans and even Rory Storm and the Hurricanes were performing Country & Western songs in their setlist.