Can a number "contain itself"?

Does pi contain pi?

No. If pi contained pi, at the point where the "contained pi" started, pi would become a repeating decimal. Then pi would not be irrational. Hence, pi cannot contain pi.

Is there any way that a number can "contain itself" other than it having a pattern that repeats itself?

Here's another way to look at it.

If a number contained itself starting after n digits, then the first n digits would be the same as the second n digits. Similarly, the second n digits of the copy would be the same as the first n digits of the copy which means, since the second n digits of the copy are the third n digits of the original number, the third n digits of the original number would be the same as the first n digits. And so on, and so on, to infinity. The number would be a repeating decimal of period n.