Aerosmith released several power ballads before recording "I Don't Want to Miss a Thing" for the soundtrack for Armageddon, but these earlier attempts retained more of Aerosmith's hard rocking instrumentation.
Steven Tyler's harmonica and Joey Kramer's drums drive the opening bars of "Crazy," while Joe Perry's fills flesh out the track. Perry and Tom Hamilton launch "Crying" with a guitar-bass power riff. On "Dream On," Perry only holds back so long before unleashing some rock and roll leads.
With "I Don't Want to Miss a Thing," though, the band's instruments are all but buried beneath the lush orchestral instrumentation. Kramer's drums provide a rock-like beat, and occasionally a guitar chord and fill peek through, but for the most part, the vocals are framed in the leanest sections by a piano and in the densest by a full orchestra.
If the band members are uncomfortable with their roles in the song, they haven't said so. For example, Perry concedes that "I Don't Want to Miss a Thing" was largely written and arranged by others.
He said, "We were out on the road, so they brought us in to see the movie and said 'Here's the song, this is where it fits into the movie, you can do it if you want.'" If they had been less busy, he says, they would have liked to write their own song, but “the song was great, people loved it, and I don't think people care that much who wrote it." (Source)