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Pretty much anything sung by George Strait is Awesome Music. There are very few exceptions and even those exceptions are somewhat awesome.
Speaking of Georges, there's also "He Stopped Loving Her Today" by George Jones, which often tops polls as the best country song ever.
Johnny Cash. "Ring of Fire" is his most well-known, but some other gems can be found: "The Man Comes Around" and his cover of "Sam Hall" from American IV are amongst the least tragic on the album, while his cover of "Danny Boy" can move you to tears.
When Johnny Cash covered NIN's "Hurt", Trent Reznor said this:
I pop the video in, and wow... Tears welling, silence, goose-bumps... Wow. [I felt like] I just lost my girlfriend, because that song isn't mine anymore.
Special mention needs to go to the sheer power and anger in "Independence Day", a song about a woman driven to the breaking point by her abusive husband, with lines like "Now I ain't sayin' it's right or it's wrong, but maybe it's the only way / Talk about your revolution - it's Independence Day," and "Let the whole world know that today is a day of reckoning / Let the weak be strong / Let the right be wrong / Roll the stone away, let the guilty pay / It's Independence Day."
"A Broken Wing." The last long note crosses over into Awesome Moments, especially live.