Define a finite set of polynomials over a field $K$ to *cover* $K$ if the images of the polynomials, viewed as functions from $K$ to itself, have union the whole set.

Define a *minimal cover* to be a finite set of polynomials that cover a field, but such that no proper subset covers that field.

Can you classify all minimal covers of $\mathbb Q_p$?

A minimal cover of a number field must consist of just a single, linear polynomial. Indeed, for most $y$, for each polynomial $f$ in the set, $f(x)-y$ is an irreducible polynomial in $x$ by Hilbert irreducibility. $y$ is in the image of some $f$ so some irreducible polynomial $f(x)-y$ has a root, so is linear, so $f(x)$ is linear. A single linear polynomial covers the set so no other polynomials are needed. This is a special case of this argument.

This does not hold in $\mathbb Q_p$. This is because $\mathbb Q_p^\times/\left(\mathbb Q_p^\times\right)^n$ is finite, so if $S$ is a set of coset representatives for $\mathbb Q_p^\times/\left(\mathbb Q_p^\times\right)^n$, then $\{ s x^n | s \in S\}$ is a finite set of nonlinear polynomials that cover $f$. It is also a minimal cover.

Is this the only kind of minimal cover of $\mathbb Q_p$, up to translation and other obvious things?

The only result I have in this direction is that if you take a cover of $\mathbb Q_p$, the leading terms of all the polynomials also cover $\mathbb Q_p$. This is because every coset of $\left(\mathbb Q_p^\times\right)^n$ (where $n$ is a multiple of the degree of each polynomial) has elements which are very large in the $p$-adic norm, and when $f(x)$ is very large in the $p$-adic norm $x$ must be very large, and when $x$ is very large, the terms other than the leading term do not change what coset $f(x)$ is in, so there must be one polynomial whose leading term can reach each coset.

A vaguely related question:

What interesting fields, other than finite extensions of $\mathbb Q_p$, admit nontrivial/interesting minimal covers?