Pi might contain all finite sets, can it also contain infinite sets?

In a previous, and quite popular, question it was discussed about whether or not $\pi$ contains all finite number combinations.

Let us assume for a moment that $\pi$ does in fact contain all finite combinations of numbers. What prevents $\pi$ from also containing all infinite sets?

It seems that at some point one would also see the first couple digits of, for example, $e$ (2.71828). But why does it need to stop there, couldn't it contain a bunch of digits of $e$? Perhaps even an infinite number of digits of $e$?

My understanding is that $e$ could also be replaced by $\sqrt2$ or any other irrational number, so long as that irrational number contained all finite sets of number combinations. Which might imply that somewhere along the way, $e$ contains a number of digits of $\pi$. Implying this ridiculous situation where within $\pi$ we see $e$, and then within $e$ we again begin to see $\pi$ again. Then all the universe collapses into a singularity. Or maybe someone can just explain why one infinite sequence can't contain another infinite sequence, and perhaps why we have not defined some type of super-infinity that can.

To reiterate the primary question: What prevents $\pi$, or other infinite irrational number that contains all finite sets of numbers, from also containing all infinite sets?

If the decimal expansion of $\pi$ contained the string of digits '$000\ldots$' then it only has a finite number of non-zero digits (some subset of the digits before the string of $0$s starts) and so $\pi$ is a rational number. Contradiction.

There are an uncountable number of infinite digit strings, and $\pi$ contains only countably many infinite digit strings-namely all the ones that are the digits after some point in the expansion. It does contain infinite strings, just not most of them.