Although I understand that, in an infinite number, (and constructed normal numbers) there could be many, many consecutive digits which are the same… the proof is in the pudding. When will they actually find a string of even a thousand identical numbers? I suspect this will never happen, even though it could. (Maybe it won’t be “bounded in a nutshell” and “call itself king of infinite space”).
It might take a while to climb up to TREE or quantify Graham crackers.
That reminds me of how the trans-uranium elements had always potentially existed, but never actually existed until created in a lab.
Algorithmic randomness implies normality. Therefore non-normality implies non-algorithmic-randomness. There are other things it can’t be also, like a “truly” random sequence that you got by flipping a coin. But at the least it can’t be algorithmically random if you can detect any bias at all in the strings of digits.
Sure, if you want the whole thing. But if you ask me what some digit is, it only takes me a finite number of steps to determine it, and the program is only of finite size.
Probably a number of them exist briefly in the aftermath of supernovae, at least up to Fermium (element 100). The extreme conditions in neutron stars could conceivably produce heavier elements but this is poorly understood.
We would have to calculate 10 to the 1000th power digits. of pi.
Sure. And I’ve seen estimates there are 10^82 atoms in the observable universe. Can you speed up the generator? In kind of a hurry here… 
The Walrus and the Carpenter
Were walking close at hand;
They wept like anything to see
Such quantities of sand:
If this were only cleared away,’
They said, it would be grand!’
If seven maids with seven mops
Swept it for half a year,
Do you suppose,’ the Walrus said,
That they could get it clear?’
I doubt it,’ said the Carpenter,
And shed a bitter tear.
With the important part here being that the program doing so only grows logarithmically with the length of the initial sequence of digits, rather than linearly, as would be the case with an algorithmically random number.
How to get the Toymaker from Doctor Who to go away and leave our universe alone: trick him into playing a “game” where he has to write out the decimal digits of TREE(3). At the rate of one per Planck time, our universe will have decayed away into random radiation, then spontaneously reformed by random chance, multiple times before he’s done.
I’ll see you and raise you A Portrait of the Artist as a Young Man:
You have often seen the sand on the seashore. How fine are its tiny grains! And how many of those tiny little grains go to make up the small handful which a child grasps in its play. Now imagine a mountain of that sand, a million miles high, reaching from the earth to the farthest heavens, and a million miles broad, extending to remotest space, and a million miles in thickness; and imagine such an enormous mass of countless particles of sand multiplied as often as there are leaves in the forest, drops of water in the mighty ocean, feathers on birds, scales on fish, hairs on animals, atoms in the vast expanse of the air: and imagine that at the end of every million years a little bird came to that mountain and carried away in its beak a tiny grain of that sand. How many millions upon millions of centuries would pass before that bird had carried away even a square foot of that mountain, how many eons upon eons of ages before it had carried away all? Yet at the end of that immense stretch of time not even one instant of eternity could be said to have ended. At the end of all those billions and trillions of years eternity would have scarcely begun. And if that mountain rose again after it had been all carried away, and if the bird came again and carried it all away again grain by grain, and if it so rose and sank as many times as there are stars in the sky, atoms in the air, drops of water in the sea, leaves on the trees, feathers upon birds, scales upon fish, hairs upon animals, at the end of all those innumerable risings and sinkings of that immeasurably vast mountain not one single instant of eternity could be said to have ended; even then, at the end of such a period, after that eon of time the mere thought of which makes our very brain reel dizzily, eternity would scarcely have begun.
Pi is an irrational number. I can’t possibly imagine it having ten in a row of any digit, including zero. I’ve spent time looking this up, and all I see are the words, “it is conjectured that”. Show me a calculation of pi that reaches a point where it has ten zeros in a row. Below is the longest calculation I could find on line in a cursory search. Looking at it, nothing seems to be remotely close to repeating ten times in a row:
3.1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679 8214808651 3282306647 0938446095 5058223172 5359408128 4811174502 8410270193 8521105559 6446229489 5493038196 4428810975 6659334461 2847564823 3786783165 2712019091 4564856692 3460348610 4543266482 1339360726 0249141273 7245870066 0631558817 4881520920 9628292540 9171536436 7892590360 0113305305 4882046652 1384146951 9415116094 3305727036 5759591953 0921861173 8193261179 3105118548 0744623799 6274956735 1885752724 8912279381 8301194912 9833673362 4406566430 8602139494 6395224737 1907021798 6094370277 0539217176 2931767523 8467481846 7669405132 0005681271 4526356082 7785771342 7577896091 7363717872 1468440901 2249534301 4654958537 1050792279 6892589235 4201995611 2129021960 8640344181 5981362977 4771309960 5187072113 4999999837 2978049951 0597317328 1609631859 5024459455 3469083026 4252230825 3344685035 2619311881 7101000313 7838752886 5875332083 8142061717 7669147303 5982534904 2875546873 1159562863 8823537875 9375195778 1857780532 1712268066 1300192787 6611195909 2164201989
It’s at position 78255277503 (and elsewhere, too, if you compute far enough and pi is normal, as is conjectured):
From here:
I found this claim on Quora.com but haven’t found verification:
As of 2015, the longest sequence found was 13 8’s at position 2.164.164,669,332 in 2.7 trillion digits of
π
.
That’s mentioned in post 11 with another cite.
Hmmm…actually, I don’t trust that site now that I’m playing with it a bit. If I keep adding zeroes, it just makes up new positions for the number, it seems, with the same surrounding numbers. There should be a better one out there somewhere. But there are ten zeroes in pi somewhere.
Here I got 8 zeroes in a row at the 172 millionth-ish position:
That seems legit. There aren’t 9 in a row in the first 2 billion digits though.
BTW: even in particle physics, let alone combinatorics, a mere 109 is nothing
It’s not a “conjecture”, e.g. ten zeros in a row occurs for the first time at the 8324296435th decimal place. You can download a file of precomputed digits, or run it yourself 
Thanks for what I assume is the accurate answer; sorry about the site I linked to above that is either buggy or full of it.
100 trillion digits here: https://pi.delivery/
Holy cow! The simple 3.1416 is plenty good enough for me. LOL