There is a paper of Shelah and Goldstern devoted to the separation of many simple cardinal invariants (this is a technical term). There are more recent papers on this subject by Kellner and Shelah, if I remember correctly.
An easy case that I am very familiar with are the so called localization numbers.
A closed set $S\subseteq\omega^\omega$ is $n$-ary if in the tree $T(S)$ of finite initial segments of elements of $S$ every node has at most $n$ immediate successors.
For $n\geq 2$ let $\ell_n$ be the least size of a family of $(n-1)$-ary sets that covers all of $n^\omega$.
Any finitely many $\ell_n$ can be separated from each other simultaneously.
This is shown in [Geschke, Kojman, Convexity numbers of closed subsets in R^n, Proc. Am. Math. Soc. 130, No. 10, 2871-2881 (2002)], which is here.
Proofs of such statements usually involve forcing with large countable support products over a model of GCH rather than iterated forcing. However, there are also some examples that use iterated forcing. See for example this paper by Shelah and Steprans.