Given that both pi and e's decimal places are completely random and infinite... I was wondering about how people say that "Every phrase ever uttered can be expressed in the digits of pi" -- which, for all intents and purposes, is true. But can e's decimal places (infinite) be expressed in pi's decimal places? Am I even asking a coherent question (like embedding infinite digits into infinite digits)?
For example, the string "27182818" is first found at position 73154827 of pi's digits; it apparently occurs 3 times in a searchable database of over 200M digits of pi. However, apparently the string "271828182" does not occur at all within the first 200M digits of pi.
Now, how about all the infinite digits of e?
Sure, limits to infinity would tell you, purely based on a probabilistic standpoint, that the chance of infinite digits (each with a probably of 1/10) occurring in a sequence of infinite digits approaches 0... but that doesn't FEEL like a satisfying analysis of the question.
Even if $\pi$ is normal, it doesn't necessarily contain infinitely long patterns. However, I think your question is ultimately unknown:
My interpretation of this is: let $\pi(i)$ be the $i$'th digit of $\pi$ $(i=0,1,2,\cdots)$ indexed so that $\pi(0)=3,\pi(1)=1,\pi(2)=4,\ldots$. Similarly, let $e(i)$ by the digits of $e$. You're asking if there exists a $k$ such that $\pi(k+n)=e(n)$ for all $n\geq 0$.
If such a $k$ exists, this means you could write $\pi=A+10^{-k}e$, where $A$ is a rational number (actually a finite digit decimal). Unfortunately, I don't think there's a known way of proving/disproving this. Specifically we don't even know if $\pi+e$ is irrational, let alone whether or not $\pi,e$ are algebraically independent.