What is the average number of factors of 2 for all numbers up to some arbitrarily large $k$ as $k$ goes to infinity?
What about factors of 3 up to $k$ ? 5?
My intuition tells me that it should be 1 factor of 2 as $k$ goes to infinity, because half of the numbers have no factor of 2, $\frac{1}{4}$ have two factors of 2, $\frac{1}{8}$ have three factors of 2, et cetera, and I think this deficit goes to zero in the limit, but I don't know how to prove it. For the others, I don't really know what the analogue to the thought process I describe above is.
You seem to be looking for this limit:
$$\lim_{k \to \infty} \dfrac{\displaystyle \sum_{n=1}^\infty \left\lfloor \dfrac{k}{p^n} \right\rfloor }{k}$$
To calculate this, it suffices to look at values of $k=p^m$ for some arbitrarily large $m$.
This gives
$$\lim_{m \to \infty} \dfrac{\displaystyle \sum_{n=0}^{m-1}p^n}{p^m} = \lim_{m \to \infty} \dfrac{\left( \dfrac{p^m-1}{p-1} \right)}{p^m} = \dfrac{1}{p-1}$$