Let $K$ be a number field and let $\mathcal{O}_K$ be its ring of integers. Since $\mathcal{O}_K$ is a Dedekind domain, every ideal has a unique factorisation into a product of prime ideals. Let $(p)$ be a prime ideal in $\Bbb Z$ and let $\mathfrak{P}_1, \dots, \mathfrak{P}_g$ be the primes lying above $(p)$ in $\mathcal{O}_K$ so that
$$(p) = \mathfrak{P}_1^{e_1}\dots\mathfrak{P}_g^{e_g}.$$
Is it possible for $e_i$ to be greater than $1$ for more than one of the $e_i$ (say $(p) = \mathfrak{P}_1^2 \mathfrak{P}_2^3$ or something) and if so, how does one describe the decomposition of that ideal in $\mathcal{O}_K$? Would one say that it is both ramified and split? If so, what is the ramification index in this case? Is it $2$ or $3$?
Yes, it is possible that $e_i>1$ for more than one index $i$. However, the ramification indices $e_i$ and the residue class degrees $f_i$ have to satisfy the degree relation $$ \sum_{i=1}^g e_if_i=[K:\mathbb{Q}]=n. $$ For $n=4$ there are examples, mentioned in the comment.