It sounds like you are talking about what in computability theory and set theory are known as **Cohen generic** reals (the lowest level of which in computability theory is **1-generic**, then 2-generic and so on).

I don't know any really natural example of a 1-generic real, but there is a fairly simple construction of one, see e.g. the book *Lerman, Degrees of Unsolvability, 1983*.

To visualize you can imagine that we put down more and more digits in our number, and every now and then we stop and say "what kind of digits could we possibly put down?", and then we put down some digits like that. For example, you could put down 100 times as many 0s as the number of digits you've put down so far. The important thing is that you eventually cover all kinds like that. So something like:

02345234

Now let's add a lot of zeroes:

0234523400000000000000000000000

Now let's add what we already have two more times:

023452340000000000000000000000002345234000000000000000000000000234523400000000000000000000000

... and so on.