A number of pro players, inlcuding Leo Kottke, prefer guitars with zero frets.
Advantages are that the open strings sound the same as the fretted strings, whereas the open strings sound different at the nut. (Though you may have to have Leo Kottke's ears to tell the difference.)
The other advantage is that the string height is guaranteed to be consistent right dowen to the zero fret, whereas the string height is harder to set at the nut.
If you've ever noticed that notes you fret at the first fret sound slightly off, intonation wise, from the notes of the open strings, it's probably because the string height is off at the nut.
Even the pro builders get this wrong sometimes. I had a Taylor 12-string that drove me crazy because of this. I was constantly retuning. Finally, in exasperation, I capoed the guitar at the first fret, retuned, and voila! -- problem gone. That was how I first learned about the importance of getting the nut cut properly. I took it into my repair guy, who replaced the nut, and said, half kidding (I think), "Yeah, we wouldn't have this problem if we went back to the days of zero frets."
But to hold up, they need to be made of stainless steel as GAD notes.
Kottke supposedly tried to get Taylor to make their Kottke signature model (LKSM) guitars with zero frets -- Bob Taylor said no.