Why the obsession with the lowest octave

A couple of responses mention lower overtones. This is not correct. An overtone is a frequency above and some multiple of a fundamental frequency. Thus the first couple of overtones for the 42 Hz of an open E string on an acoustic bass are 84 and 168 Hz. There is no 21 Hz overtone.

http://en.wikipedia.org/wiki/Overtone

From demonstrations I've heard, the most important benefit of a subwoofer (or bi-amped full range speaker if the bass crossover is low enough) is improved clarity and definition in the upper-bass, lower-midrange. I think this results from removal of the lowest source frequencies (requiring the greatest energy) from an upper bass driver and the main amplifier. I've found this case no matter what type (frequency range) of music is played.

I've found it interesting that while the OP ask about the "lowest octave", which would commonly be defined as 20-40 Hz, so many replies have commented on bass in general or if specific, reproduction above 40 Hz.

But for the general value of a decent subwoofer (or bass extension), I had a learning experience similar to Ecruz. Many years ago I visited a well-respected dealer for a sub-woofer demo. The main speakers he chose were Spendor BC-1s (that may hint how long ago this happened) with an excellent mid-range reputation but known for low frequency limitations. But to my surprise, for the music demo he used a solo violin. Even with such a limited low frequency range of the instrument, it was very obvious when he switched the sub system in and out. That forever sold me on the value of extended bass (again in general, I'm not identifying a specific frequency) for a fuller musical experience.