Bekenstein bound for electron? Using the Wikipedia version of the Bekenstein bound, and substituting the Wikipedia values for electron mass and radius, one obtains 0.0662 bits. Does this really mean that a system, any system, placed inside a sphere the size of an electron, and weighing no more than an electron does, is almost determinate? How about an electron itself? Wouldn't one need at least a few bits to characterise the behavior of an electron in magnetic space? 
(I am a professional mathematician but I know very little about physics, I'm sure I'm missing something obvious here...)  
 A: You have found an elaborate way of calculating $2\pi \alpha/ \ln 2 \approx 0.0661658$. Here, $\alpha \approx 1/137$ represents the fine-structure constant.
The points to note is that:
A) Bekenstein's bound defines the maximum number of nats of information that can be contained in a spherical region as the circumference of that region divided by the reduced Compton wavelength associated with the total energy contained within that region, 
and 
B) the classical electron radius is equal to the fine structure constant times the reduced Compton wavelength of the electron. 
Would you redo your calculation using the electron mass and the reduced Compton wavelength of the electron, you would obtain a value of $9.0647$ bits. However, you would obtain exactly the same value for a proton or whatever other elementary or composite particle you might chose. I would not attach any physical significance to these results.

Added: We currently don't have a consistent quantum gravity theory, and we don't even have an idea what would be the fundamental degrees of freedom in such a theory. Therefore any statement in response to questions like "how many bits/nats of information can be associated with an electron mass" runs the risk of leading to nonsense. Having said this, the holographic (Bekenstein-Hawking / black hole) bound seems more capable of providing reasonable leads. Using $4\pi$ times the square of the reduced Compton wavelength of the electron as area in the BH bound leads to an information content of $S/k = \pi \hbar c /G m^2$ nats. Here $m$ denotes the electron mass. This result for "the information content of a volume large enough to contain an electron" is in essence the square of the ratio of the Planck mass over the electron mass. That's a lot of nats. 
A: One can't take results like that too seriously at the scale at which an electron would apply.  In particular, the classical general relativistic model, applied naively to a point mass electron would tell you that the electron has too large a charge and angular momentum to have a black hole horizon, and would instead be the exotic type of object called a naked singularity.  
