This is great input and I will be looking into this in detail! (By the way I am currently trying to narrow down my interests and this is exactly the kind of feed back that I was looking for). However, a quick look into this tells me that it is expected (conjectured by who?) that $e$ and $\frac{1}{\pi}$ are not in fact periods. I have edited my original question to include more background, and while I feel like periods would be examples of the numbers I am considering, I may actually be thinking of a larger collection.