Yes, this is true, there are real numbers that are "more complicated" than the set of real numbers.
But it's quite unfair to say that this is a problem with real numbers only. The situation truly shows up everywhere.
For example, the set of prime numbers are very easy to describe, but the seperate prime numbers can be quite complicated.
This is just one example, but the more you progress in mathematics, the more that the situation shows up. It's generally easier to describe the set of elements than the individual elements.