He offers some here:

When I teach a course, I too like to sell the sizzle at the beginning of each lecture. For a graduate course in information theory I teach, the students are told that they will learn why their cell phones use recently discovered coding that pushes the boundaries of what is mathematically possible in communication speed. I also tell them that we will prove that some things exist that we can also prove are unknowable. And there are numbers that a computer can’t compute. There also exists a single number, Chaitin’s number, that we know lies between zero and one. If we knew Chaitin’s number to finite precision, we could prove or disprove numerous open problems in mathematics. Large monetary awards await anyone who supplies a proof.

But guess what? Even though we know that Chaitin’s number exists, we can also prove it is unknowable. How’s that for sizzle? Contra-nerds may be bored. STEM nerds will drool with anticipation. I am convinced sizzle can be identified for every STEM class. I teach a number-crunching class on multidimensional signal processing where we talk of iterative algorithms converging to a fixed point. Sounds pretty boring huh? But consider the following sizzling example, understandable to all: …

Robert J. Marks, “STEM Education 7: Sell the Sizzle” atMind Matters

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*See also:* Top Ten AI hypes of 2018

Chaitin’s number(s) (they comprise an infinite family) are indeed fascinating. However, the first (leftmost) several digits of some Chaitin’s numbers are known (source). I haven’t read the papers, but for example one such Chaitin number begins 0.00787499699… in decimal form.

One surprise was the discovery of the number between 6 and 7 they called “bleen”. (RiP GC) 😎

Regarding this:

No doubt that the various Chaitin’s numbers are not computable. But unknowable? That’s a separate question, I believe.