First, a few links:
You can't toot your own horn if you only have an OBOE.
I would not at all be surprised if there were offbyone errors (OBOE) in some of the positions I reported in pi and tau. When programming, I do zerobased counting, but I tried to express position numbers with more familiar onebased counting ordinals. A further complication is that it seems to be traditional to count the digits of pi starting after the decimal point, ignoring the three. Either these sources of error will cancel out or I'll actually have Off By Two Error, I don't know anymore, my head hurts. I tell you, Off By One Error are the two most annoying things about programming.
You keep using that word, "random". I do not think it means what you think it means.
It is very problematic to use the word random in relation to pi: there's absolutely nothing random about it, it's a constant, every digit is perfectly predictable if you have the time and patience and computing ability. I know what you're going to say, "but it's random in that its digits don't, aren't..." and I'll just stop you right there, I thought that too, but I'm told on good authority in no uncertain terms that that's not what random means. Mathematicians call the behavior of the digits of pi, or tau, or e, or √2, etc. that of
normal numbers, e.g. the longer you count the digits, the more equal the count of every single digit. There are two problems with this nomenclature: one, the word "normal" is abnormal in that it means a lot of different things even within one discipline such as mathematics (normal numbers have little to do with normal distributions), so it's prone to be misunderstood, and (2) there is no proof that pi is normal, it just looks really, really, really normal every time someone carries the calculation a little farther.
tl;dr: I'm calling pi
randomesque. It's totally meaningless, but instantly grokkable, I think.
It's hip to R Squared
I was unsure whether to use the term R squared, since it's so often used in a context of regression analysis. But I went with the familiar term rather than Coefficient of Determination. Just remember, the R Squared is not comparing the distribution to its best fit regression line, but to the line of a uniform distribution (with a slope of zero). R squared is most often used to analyze a regression model, but it's perfectly valid to use the term to compare a distribution to any model.
Home on the range
I found this somewhat interesting; there are plateaus and long reversals in the range between the most common and least common digits in the cumulative count.
The stats, ma'am, just the stats
Here are the complete stats mentioned in passing in part two of the competition:
NOTE: Some of the Tau ones might be wrong; a courageous reader found a bug in my program. I think the pi ones are correct, though, and any errors in Tau would make Tau better in comparison to Pi, so it would not change the results of my totally arbitrary contest, thank goodness.

pi 

tau 

length 
position 

length 
position 
Consecutive even numbers 
29 
36,454,143 

28 
39,904,078 
Consecutive odd numbers 
30 
92,438,125 

28 
2,946,687 
Consecutive prime numbers 
26 
896,631,791 

22 
84,259,349 
Consecutive binary numbers 
8 
42,408,101 

8 
65,607,193 
Longest stretch lacking one number 
196 
18,522,937 

210 
362,783,626 
Recapitulation of itself 
9 
50,366,471 

10 
19,683,238 
Recapitulation of the other 
10 
19,683,238 

11 
52,567,169 
Number nine, number nine
When talking about R squared, a common question is "how many nines?", i.e. is the statistic above 0.9 (one nine), 0.99 (two nines), 0.999 (three nines), etc. The R Squared of Pi's cumulative digit distribution vs. a uniform distribution hits seven nines at position 72,000,000; I did not check past 100 million. The pattern seems to be approximately adding a nine with every tenfold change in position. Maybe that's some sort of inevitability with nearuniform distributions, it sounds likely, if any mathematicians read this and are like, "well, duh," let me know.
R squared 
position 
0.9 
19 
0.99 
577 
0.999 
6,410 
0.9999 
55,049 
0.99999 
461,828 
0.999999 
4,840,626 
0.9999999 
72,819,444 
A final thought
It wasn't long ago that it was difficult to calculate pi. Now I could have downloaded a program to generate hundreds and hundreds of millions of digits on my desktop computer in a reasonable amount of time. Maybe the fact that supercomputers have pushed our knowledge into the hundreds of billions of digits makes it less fascinating, but many of the geekout pi pages on the Internet date from the '90s.