Notation for "the highest power of $p$ that divides $n$"

Yes, there is a standard notation, namely $p^e\mid\mid n$, which says that $e$ is the largest power of $p$ which divides $n$.

Reference: Martin Aigner, Number Theory.

Edit: For more advanced purposes, like $p$-adic numbers etc., a common notation is also $\nu_p(n)$, which also then appears in more elementary context. For elementary number theory I have seen $p^e\mid\mid n$ more often, though.

This can be seen as a special case of the more general concept of valuations (on discrete valuation rings).

A common notation in that context, which is quite convenient also here is $\nu_p(n)$.