Forum:Questions in googology

Hello, I am a mathematics student at the University of California – Los Angeles. I have always had a soft spot in my heart for googology and so I have begun to write a rigorous introduction to the main concepts. My hope is to come up with something that will both motivate its study to the mathematically trained – who generally tend to regard it with disdain – and also suggest new directions of research for those it catches on.

But I need questions. This request is twofold: I need questions I can answer with proofs in order to demonstrate the existence of interesting googology-related mathematics, and I need to provide questions for further research. So I am asking all of you: what googology-related questions would you like answered? They can be philosophical or rigorous questions, as long as they would be interesting to a mathematically trained individual. Also, I know it can be hard to classify a question as "interesting" or not, so just err on giving me your question if you are unsure. Finally, don't hesitate to give me multiple questions or ask more about my background/this project!

Examples of questions:
 * Do busy beavers have strongly connected graphs?
 * Can we come up with a "reasonable" logic stronger than set theory that still allows us to formalize the idea of a number and extract large numbers out of it?
 * Is there a fast algorithm which, given two numbers in Conway Chained Arrow Notation, returns which one is larger (or if they're equal). See for example https://mathoverflow.net/questions/119453/polynomial-time-algorithm-to-compare-numbers-in-conway-chained-arrow-notation.
 * How quickly does the representation of an ordinal increase if we take an element from its fundamental sequence? For example, upon taking the 3rd element of its fundamental hierarchy the ordinal ω3becomes ω2×3, and when we do this again we get ω2×2+ω×3. If you try this with random ordinals and take the kth element of the fundamental sequence it seems they never increase in size by more than a factor of k. Can we prove this?
 * For certain choices of "reasonable" ordinal representation systems (say, Cantor normal form, which works up to ε_0), does the busy beaver function "effectively" beat the fast growing hierarchy for all ordinals in this representation system? That is, is the lowest number N for which BB(n)>f_α(n) for all n>N at most f_α(C) for some reasonably small value of C? The motivation for this is that if not, then of course BB(n) still dominates f_α(n), but it's not practical for naming larger numbers than f_α(n) since the size of numbers we must plug in to get a bigger output from BB(n) is too large for us to express.