How thick is the cosmic microwave background, including the part we cannot see within the observable universe? What I want to know is how thick the observable universe is from the point of the cosmic microwave background and beyond.

It appears the thickness of the cosmic microwave background itself (the part we can see) is above 100,000 light years, per the following article:
http://scienceblogs.com/startswithabang/2013/06/19/5-facts-you-probably-dont-know-about-the-cosmic-microwave-background/

However, I want to know the thickness of that, plus what lies beyond that we cannot see, another way of looking at it would be the distance between the surface of last scattering (cosmic microwave background end) to the beginning (e.g. Big Bang).

According to the following article, it appears that this time from the beginning to the surface of last scattering is about 300,000 years:
https://ned.ipac.caltech.edu/level5/Glossary/Essay_lss.html

That would imply that the thickness should be about 300,000 light years, but that doesn't take inflation into account. 

What is the thickness (in the observable universe) between the beginning (e.g. Big Bang) to the surface of last scattering (Cosmic Microwave Background), including inflation?
If I understand you correctly, you want to know the distance from the point from which we observe the CMB, to the edge of the observable Universe.

During inflation, the observable Universe expanded from ridiculously small to some ten meters in radius, so that part can be safely ignored compared to the distances now$^1$.

The distance$^2$ to the "CMB shell"$^3$ is 45.4 billion lightyears ("Glyr"), and the distance to the edge of the observable Universe is 46.3 Glyr. Hence, the shell of the observable Universe that lies beyond the CMB shell has a thickness of only 0.9 Glyr.

Here's a sketch of how I interpret your question (not to scale):



If you want, I can add details on how to calculate these numbers.



$^1$The relative expansion during inflation was huge, however: The Universe expanded roughly by the same factor that it has expanded afterwards, namely a factor of $\sim10^{26}$.

$^2$Here, "distance" corresponds to the comoving distance, which is what you would measure if you froze the Universe right now, and started laying out meter sticks.

$^3$This shell is not infinitely thin, but has a thickness of some 60 million lightyears, so let's ignore that.