I found this link about frequencies of digits appearing in π : http://www.eveandersson.com/pi/precalculated-frequencies
This made me wonder: do those frequencies ever become even on the lifespan of the currently known digits? (each of the 10 digits [0,1,...,9] appearing n times on the (10*n)'th digit)
If not, what's the closest they get to, and when?
It's unlikely. If you take $10n$ random decimal digits, the probability that you have $n$ each of $0, 1, \ldots, 9$ is $(10 n)! 10^{-10n}/(n!)^{10}$. In an infinite sequence of random digits, the expected number of positive integers $n$ for which this is true is $$ \sum_{n=1}^\infty \frac{(10 n)! 10^{-10n}}{(n!)^{10}} \approx 0.0003933209904$$ most of which ($0.00036288$) comes from the $n=1$ term.
Of course the digits of $\pi$ are not really random, but their statistical properties are (as far as we can tell) similar to a random sequence, and in this case we know that the first $10$ digits are not evenly distributed (e.g. digits $2$ and $4$ are both $1$).