If $K$ has characteristic $p>0$, an
[Artin-Schreier extension][1]
$L/K$ is cyclic of degree $p$, and is generated by an element $a \in L$ such that $a^p \not \in K$. Such
an $L$  has the form $K(a)$ for a root $a$ of a polynomial of the 
form $X^p - X - \alpha$ for suitable $\alpha \in K$; the polynomial
then splits over $L$ since its roots are $a+i$ for $i \in \mathbf{F}_p$
(in this way you can see the action of the Galois group $\mathbf{Z}/p$ on the roots).

An example of such an extension is $\mathbf{F}_4/\mathbf{F}_2$ since $\mathbf{F}_4 = \mathbf{F}_2(a)$ for a root $a$ of $X^2 - X - 1 = X^2 + X +1$.


**Edit(x2):**
Looking at Georges Elencwajg's answer, I see that I made what he labels the "First Interpretation" of the question. Since his "Second Interpetation" seems interesting, here is what I hope is an example.

Let $k$ be a field of char $p>0$, and let
$K = k(t)$ for transcendental $t$. Let $L/K$ be the Artin-Schreier extension obtained by adjoining to $K$ a root $a$ of $X^p - X - t$.


I claim that $a^N \not \in K$ for any $N \ge 1$.

Well, note that $k[a,t]$ is an integral extension of $k[t]$. Hence if $a^N \in K$,
then in fact $a^N \in \langle t \rangle \subset k[t]$, so that $a$ satisfies a polynomial
$$X^N - tg(t) \quad \text{for some $g(t) \in k[t]$}.$$
But then $X^p - X - t$ divides $X^N - tg(t)$ and reducing mod $t$ 
gives a contradiction. (i.e. the condition $a^N \in K$ would mean that $L/K$ should be
ramified at the prime $\langle t \rangle$ of $k[t]$, which isn't so).

So: this cyclic extension is generated by an element $a$ for which $a^N \not \in K$
for any $N \ge 1$. I expect (?!) that this should give an example for Elencwajg's "Second Interpretation" of the question, but I didn't think about other possible generators
for the extension $L/K$.


  [1]: http://en.wikipedia.org/wiki/Artin%25E2%2580%2593Schreier_theory