Take a look at 0.101001000100001... This number is also non-repeating, but obviously doesn't contain all numbers with finite digits.

The property you're looking for is called to be a normal number. Pi is assumed to be one, but it hasn't yet been proven.

However, in a sense this is an unremarkable property as almost all real numbers are normal. :)

At work at the moment so can't go deep into it. But I think you misunderstand what non repeating numbers mean. Of course there are repeating numbers within pi which is fine, the issue would be if ALL the digits were to simply cycle over and repeat themselves. If however there are a few trillion digits then a series of 1's and 0's for ever, pi is still non repeating