Take any polynomial of degree $e$ that is Eisenstein at $p$, adjoin to $\mathbb{Q}_p$ a root of that polynomial and you will get a totally ramified extension of $\mathbb{Q}_p$ of degree $e$. Moreover, all totally ramified extensions of local fields arise in this way, by adjoining a root of an Eisenstein polynomial. So the ring of integers of that extension will be what you are looking for.
Edit: I should have said that all the DVRs you obtain with the above procedure are of the form $\mathbb{Z}_p[\pi]$, where $\pi$ is a root of the Eisenstein polynomial you started with. See e.g. Serre, Local Fields, p 58.
Take any polynomial of degree $e$ that is Eisenstein at $p$, adjoin to $\mathbb{Q}_p$ a root of that polynomial and you will get a totally ramified extension of $\mathbb{Q}_p$ of degree $e$. Moreover, all totally ramified extensions of local fields arise in this way, by adjoining a root of an Eisenstein polynomial. So the ring of integers of that extension will be what you are looking for.