Possible, yes. Recommended, probably not.
As other answers have stated, it is certainly possible to write a song that does not conform to defined tonal "scales" like the Ionian mode (major scale) and related modes, pentatonic scales, and variations like harmonic and melodic minor (which differ from "natural minor", the Aeolian mode, in a couple subtle ways such as a major 6th or 7th).
If a melodic line does not generally conform to one of the above, it is generally termed "atonal". However, it can still conform to a system; music written in the whole tone or tritone scales, for instance, or in Middle Eastern and Indian systems which utilize quarter-tone differences. Whole tone and tritone music can still be written in Western notation, but many Eastern styles cannot.
Then, there are music styles that defy printed form; many traditional African call-and-answer styles were passed down for thousands of years solely through word of mouth, and completely lack a tonal center, focusing instead on rhythm and flow of the song. With the sweeping popularity of Western culture and musical instruments, particularly keyboards and synthesizers, a lot of African music has become more systematic, but you still hear very odd intervals reminiscent of the traditional songs.
In all cases, the Western music notation system evolved to handle Western music styles, which use the 12-tone system, key and time signatures. Even then, some contemporary music styles, especially a capella choral music and jazz, can make a pianist cringe when told to decipher it in condensed staff form.