The first example normal number that has been produced. It was discovered by David Champernowne at Cambridge University, a friend of Alan Turing's. It goes:
More precisely, write down a decimal point, then write the numbers 0,1,2,3,4,5,6,7,8,9 then 10,11,12,13,14,15..., till you reach 99, then you write 100,101,... and go on this way forever.
Champernowne showed it is a normal number in base 10 and only in base 10. In base 10, each possible digit from 0-9 occurs exactly 10% of the time in the limit, each two-digit block exactly 1% of the time, each three-digit block 0.1% of the time, and so forth.
It's still an open question as to what happens to Champernowne's number when it is expressed in other bases.