If we have two primes, $p$ and $p'$ and the $p,p'$-adic integers $\mathbb{Z}_p$ and $\mathbb{Z}_{p'}$. Then the integers inject into both of the rings, so we can say that $\mathbb{Z} \subset \mathbb{Z}_p \cap \mathbb{Z}_{p'}$. My question is, are the integers the entire intersection? I assume they are as all the other elements of $\mathbb{Z}_p$ and $\mathbb{Z}_{p'}$ rely on what $p$ and $p'$ are.
The comments have determined that $\mathbb{Z}$ is not the largest ring that embeds into both $\mathbb{Z}_p$ and $\mathbb{Z}_{p'}$. But is $\mathbb{Z}$ the largest ring that injects to all $\mathbb{Z}_p$.
I think this is a case where you need to distinguish between "injects into" and "is a subset of". Each of $\mathbb Z_p$ and $\mathbb Z_{p'}$ is an abstract thing that one can choose to realize formally in several different ways (which we don't normally distinguish between because they produce isomorphic results) -- and which precise formalism we use will determine whether the sets that represent $\mathbb Z_p$ and $\mathbb Z_{p'}$ even intersect at all. Whether they do is more a property of the "uninteresting" choices you make during the construction, than it is a property of the idea of $p$-adic integers per se.
For example, you may choose to realize $\mathbb Z_p$ as a set of sequences of digits $0\le d<p$, and in that case there will certainly be an intersection, such as the number $...1111$. But it will behave differently under the $p$-adic arithmetic than under the $p'$-adic arithmetic.
A better question might be whether there's a larger ring than $\mathbb Z$ that embeds homomorphically into both of $\mathbb Z_p$ and $\mathbb Z_{p'}$.
The answer to that is yes. For example, unless I'm mistaken $\mathbb Z_p$ always contains a subring isomorphic to $\mathbb Z[X]$, with $X$ represented by some element of $\mathbb Z_p$ that is transcendental over $\mathbb Z$. I think that $\sum_{k=0}^\infty p^{f(k)}$ where $f$ is a "sufficiently fast-growing" function will qualify, but even if not, a cardinality argument says that there must be some transcendentals in $\mathbb Z_p$ that can be used.
There's no canonical identification of these copies of $\mathbb Z[X]$, of course.
Note that this also makes $\mathbb Z[X]$ a ring that embeds into all $\mathbb Z_p$s. (It's not the largest such one, of course; you can add continuum many different unknowns and still embed into all of them, given enough choice).