This is sort of a specious question, a little like "how many atoms does it take to make a person?"
First off, the spherical wavefront image is not accurate in the real world. That's what a theoretically perfect wave would "look" like if the medium it traveled in was perfectly still and the source of the disturbance was a point or a perfect sphere itself (like a perfectly-shaped ball suddenly expanding). I reality the media (the air, the walls, etc.) are different, the air's in motion and has other sounds already moving around in it too; the guitar string is not a point source but a long line; much of what you actually hear is the part of the string vibration resonating in the body of the guitar, resonating, and projecting out the the sound hole, diffracting at the edges of the hole and off the strings themselves...
Further, one pluck sets the string to vibrating, and then it stays vibrating for a long time if you don't pluck it again and just let it ring. So the sound waves created begina and gradually fall off. When do you say the string has stopped vibrating and there are no more sound waves being produced? When you can't hear it any more? When a precies measurement device says the string has stilled?
Finally, the concept of "a" sound wave is merely a conceptual model. What you hear is due to the motion of the air striking your ears. This is the sum of all the motions of all
the disturbances going on at one time. If you could see it it would look like a random mush of air molecules moving about. There is no way to separate the part of the total disturbance caused by the guitar from all the rest -- there are not a bunch of physical spherical wavefronts you can see individually and say, "ah, there's one of the guitar waves..."
All in all, you can only do this sort of counting in a theoretical imagined state that does not exist in the real world.
So the real answer is -- it's not possible to know how "many waves" there are.
If for some reason you're curious about how many waves there might be from a perfect point source in a perfectly still and homogenous medium, the first answer tells you all you need to know. The point source would have a single frequency of vibration and no harmonics, so the frequency tells you how many wave fronts pass out from the source per second. Then you multiply the frequency by the how long the source vibrates.
Voila -- x number of waves, a totally imaginary and useless number except for it value as a mental exercise.
