For me, it’s Hank Williams - I’m So Lonesome I Could Cry
BJ Thomas had a good version, but Hank’s is more direct and more effective for me.
“Blue Eyes Cryin’ in the Rain” by Willie Nelson was the first thing I thought of. I suppose something by Patsy Cline (“She’s Got You” maybe) would have to be high on my list.
Boulder to Birmingham by Emmylou Harris. I could list many, many more though.
Oh, my, there are so many!
If I had to pick just one, though, I believe I’d have to go with “Sing Me Back Home”, by Merle Haggard.
I only know relatively recent stuff, not classic country, but I can’t stop loving “Heads Carolina, Tails California” by Jo Dee Messina. It just totally makes me want to drop everything and hit the road for an adventurous, unplanned vacation.
Either Long Gone Lonesome Blues or I Gotta Get Drunk.
Johnny Cash’s Ring of Fire, though El Paso by Marty Robbins gets honorable mention.
Not a problem here. I almost used Top 5 or Top 10 in the title, but went with Favorite so as not to scare folks off!
Other oldies I could list:
San Antonio Rose
Tennessee Stud
Wildwood Flower
Sixteen Tons
Steel Guitar Rag
You Never Even Called Me By My Name
David Alan Coe
I’m So Lonesome I Could Cry is also the first song I thought of. But I’m sure there will be a dozen others suggested that I like about as well.
Blue Eyes Crying In the Rain is just about there.
Yesterday I was listening to Elvis Costello do Good Year for the Roses (most famously performed by George Jones). If the lyrics of the second verse were as good as the first, that might be my favorite.
First verse and chorus excerpt:
*I can hardly bear the sight of lipstick
on the cigarettes there in the ashtray,
Lying cold the way you left them
but at least your lips caressed
them while you packed.
And a lip-print on a half-filled cup of coffee
that you poured and didn’t drink,
but at least you thought you wanted it
that’s so much more than I can say for me
But what a good year for the roses…
*
Another country song that revisits my mind every few months is Haggard’s “If We Can Make It Through December”.
It’s always tough to pick a single favorite anything, but I’ve always thought (despite what David Allan Coe says) that You’re The Reason God Made Oklahoma is just about the perfect Country song.
(If you read the lyrics without knowing the song, it’s a duet. He’s in Tulsa and she’s in L.A. I can’t link to youtube from work, sorry.)
Wichita Lineman sends shivers up my spine.
Seven Spanish Angels is a favorite. I’m also a big fan of Lefty Frizzell - I Love You A Thousand Ways.
I don’t know if it qualifies, but the first song that came to mind was Dead Flowers.
Oh oh oh I forgot Mary Chapin Carpenter’s John Doe No. 24. The most moving thing I’ve ever heard.
Well since I have permission to list a few,
He Stopped Loving Her Today by George Jones has to be mentioned, in fact I’m surprised no one has so far.
Others I love include Return of the Grievous Angel by Gram Parsons, Over Yonder (Jonathon’s Song) by Steve Earle, The Angel’s Rejoiced Last Night by The Louvin Brothers, and The Purgatory Line by Drive-By Truckers.
I could go on for quite some time, as a youngish guy in England I don’t get much chance to go on about country music!
Carry on, Friend! No reason not to explore this genre to its depths and boundaries – if they exist.
I’m truly impressed with the seriousness this thread is greeted with. To see this many responses this quickly is amazing to me!
More!
Zoe wanted to know if I was including “Western” in the “Country” classification, and I told her I hadn’t put any qualifications or limits on the idea of “Country” so she said hers would have to be
Ghost Riders In The Sky
My favorite version is by Vaughn Monroe because his was the first version I heard as a little kid. Maybe she’ll say whose version she prefers. Johnny Cash’s is famous, for sure.
One of the first songs I remember loving as a child was Johnny Cash’s version of The Long Black Veil, which probably tells you something about me… More folk than country perhaps, but in the same general area.
Recently I’ve been listening to more bluegrass, and one song that comes to mind is Shady Grove by Bill Monroe. That’s one to play when friends of mine who’re into extreme metal think there’s no speed or technical playing in country music 
One last one (for now), Ralph Stanley’s version of O Death from the O Brother Where Art Thou soundtrack, and especially the performance at the Grammy awards where it won so many awards. Spellbinding.
Strange, but Cash’s version of NIN’s Hurt is one of my favorites.
I also like the goofy country tunes, like Red Necks, White Socks and Blue Ribbon Beer.